Catherine Shine filed this action in May 1987 against numerous manufacturers and distributors of asbestos, including the appellants herein, to recover damages allegedly resulting from her husband’s exposure to products containing asbestos. Mr. Shine died of lung cancer in November 1986.
*95This case was originally consolidated with other asbestos cases before Judge Charles Kocoras for pleading, discovery, and pretrial purposes. On October 27, 1986, Judge Kocoras entered Consolidated Pretrial Discovery Order No. 1 (PTO 1), which provided that the plaintiffs in these cases disclose product identification information within one year of the filing of their respective complaints. Under the terms of PTO 1, Shine was required to identify by May 22, 1988, any individuals that would testify at trial regarding the asbestos-containing products to which Cornelius Shine had allegedly been exposed. PTO 1 also required Shine to provide other information, including any specific asbestos-containing products the witness planned to identify, the defendant to which the witness attributes such products, and approximate dates and specific locations where the products allegedly were seen or used. On May 18, 1988, Shine served the defendants with a “disclosure” listing four witnesses, but neglected to provide information regarding specific products, dates, or locations.
Judge Kocoras subséquently entered PTO 2, which initiated a “spin out” process whereby five cases would leave the consolidated pre-trial asbestos docket each month and return to the assigned trial judge for disposition. The “spin out” dates also served as a means for calculating deadlines for discovery obligations. Under the terms of PTO 2, Shine was required to provide the product identification information no later than eight months before her “spin out” date, which was May 15, 1991. Therefore, Shine had until September 15,1990, to provide the information specified in PTO 1. Shine failed to comply with PTO 2, providing no witnesses by that deadline date.
Judge Kocoras thereafter entered his final order, PTO 3, which established May 1, 1990, as the last date on which Shine could supply the defendants with co-worker “pre-deposition disclosures.” Significantly, PTO 3 also provided that “no co-worker will be permitted to testify at the deposition or at trial concerning a defendant, a product, a time period or a location not identified in the predeposition disclosure, absent a showing of extraordinary circumstances made by written motion and supported by an affidavit attesting to the truth of the fact set forth therein.” Once again, despite this admonition, Shine failed to produce requisite disclosures.
Judge Suzanne Conlon, the judge to whom Shine’s case was “spun out,” conducted a status hearing on March 5, 1991, at which she set April 25, 199Í, as the deadline for filing the final pretrial order, and scheduled the four-year-old case for a May trial. Although this trial date preceded the May 15 “spin-out” date established by Judge Kocoras, the plaintiff did not object. Although the defendants did object, the plaintiff stated that only a few minor matters required further attention, and the court rejected the defendants’ request to vacate the accelerated trial date.
On March 28, Owens-Corning moved for summary judgment, contending that Shine failed to produce evidence of her husband’s exposure to Owens-Corning products. Shine filed a response to Owens-Corning’s motion on April 11; with it, she disclosed, for the first time, her product identification witnesses. One week later, on April 18, Shine informed the defendants — again for the first time — that family members also would provide product identification testimony.
After concluding that Shine failed to es-. tablish her husband’s exposure to Owens-Corning products, the district court entered summary judgment in favor of Owens-Corning on April 22. In rendering its decision, the district court refused to consider Shine’s product identification witnesses, ruling the disclosures untimely in light of Judge Kocoras’ September 15, 1990, disclosure deadline, and adhering to PTO 3’s mandate excluding untimely disclosures. The court found Shine’s disclosure at such a late date “inexcusable and highly prejudicial” to Owens-Corning.
On April 23, defendants Owens-Illinois, Keene, and Owens-Corning — prior to learning of the April 22 order granting summary judgment to Owens-Corning — jointly moved to bar the plaintiff’s use of the *96belated disclosures. Although the court initially denied the motion as moot on procedural grounds not relevant here, Owens-Illinois and Keene subsequently renewed the motion. Shortly thereafter, Owens-Illinois and Keene filed a joint motion for summary judgment, alleging that no competent evidence existed to establish that Mr. Shine’s injuries were.caused by their products. The district court granted the motion to bar the product identification witnesses on May 16, for virtually the same reasons as those delineated in its April 22 order, and on June 15, granted summary judgment to Owens-Illinois and Keene. The central issue on appeal is whether Judge Conlon properly barred Shine’s product identification witnesses under Fed. R.Civ.P. 37.
We review the district court's decision to impose Rule 37 sanctions for abuse of discretion. Parker v. Freightliner Corp., 940 F.2d 1019, 1024 (7th Cir.1991); Sere v. Board of Trustees of Univ. of Ill., 852 F.2d 285, 289 (7th Cir.1988); Tamari v. Bache & Co. (Lebanon) S.A.L., 729 F.2d 469, 472 (7th Cir.1984). Indeed, because the district court is in the best position to determine whether a party has complied with discovery orders, its discretion “is especially broad.” Parker, 940 F.2d at 1024; see also Patterson v. Coca-Cola Bottling Co., 852 F.2d 280, 283 (7th Cir.1988) (trial court’s sanctions will not be set aside unless no reasonable person could agree with its assessment of issue under consideration); 3 Penny Theater Corp. v. Plitt Theatres, Inc., 812 F.2d 337, 339 (7th Cir.1987) (same). As we previously have observed, “judges must be able to enforce deadlines.” Parker, 940 F.2d at 1024 (quoting In re Kilgus, 811 F.2d 1112, 1118 (7th Cir.1987)).
Here, we find that the district court did not abuse its discretion in disallowing the untimely evidence produced in contravention of the deadlines set by Judge Ko-coras’s pre-trial orders. Indeed, Judge Ko-coras, in an opinion rendered subsequent to Judge Conlon’s disposition here, relied on her reasoning and likewise found a plaintiff’s failure to disclose product identification witnesses as ordered by PTOs 2 and 3 “unexplained and inexcusable.” Siwy v. Keene Corp., No. 87 C 6146, 1991 WL 139996 at *2, 1991 U.S.Dist. Lexis 10074 at *5 (N.D.Ill. July 22, 1991). Following Judge Conlon’s approach in the instant case, Judge Kocoras refused to consider the expected testimony and granted the defendant’s motion for summary judgment. Id. 1991 WL 139996 at *2, 1991 U.S.Dist. Lexis 10074 at *5-6; see also Fitzgerald v. Keene Corp., No. 87 C 6139, 1991 WL 136019 at *5 n. 1, 1991 U.S.Dist. Lexis 9713 at *9 n. 1 (N.D.Ill. July 15, 1991) (Nordberg, J.) (same); Trzesniak v. Keene Corp., No. 87 C 3069, 1991 WL 60521 at *2 n. 1, 1991 U.S.Dist. Lexis 4636 at *1 n. 1 (N.D.Ill. Mar. 27, 1991) (Williams, J.). Shine further claims that the sanctions are inappropriate given the lack of notice as to their imposition. We disagree. Aside from the fact that the “district court is not required to fire a warning shot” prior to imposing sanctions, Parker, 940 F.2d at 1025 (quoting Hal Commod. Cycles Mgmt. Co. v. Kirsh, 825 F.2d 1136, 1139 (7th Cir.1987)), by their very language the PTOs provided adequate — indeed, explicit — notice that product identification witnesses not disclosed in compliance with the scheduling order would be excluded.
Shine offers a related claim, arguing that by removing the case from the consolidated group prior to the scheduled “spin-out” date, the district court “effectively cut off two-and-one-half months of discovery time and never entered either a substitute schedule or a discovery cut off date.” Appellant’s Br. at 25. This contention lacks merit. For one thing, Shine does not dispute the district court’s jurisdiction over the case or its right to remove it from the “spin-out” schedule. Id. Moreover, the district court’s decision to accelerate the schedule — to which Shine offered no objection at the March 5 status hearing— did not affect the dates set by PTOs 1, 2, and 3. That the pace was quickened did not alter the fact that Shine had already failed to comply with Judge Kocoras’s orders. Accordingly, the court acted properly in excluding information disclosed in an untimely fashion.
*97Shine also claims that, even absent the evidence excluded by Judge Conlon, sufficient evidence exists from which to glean a genuine issue of material fact sufficient to withstand summary judgment. We review de novo the district court’s grant of summary judgment, and examine all evidence in the light most favorable Shine, drawing all reasonable inferences in her favor. Anderson v. Liberty Lobby, Inc., 477 U.S. 242, 255, 106 S.Ct. 2505, 2513, 91 L.Ed.2d 202 (1986).
We turn first to the April 22, 1991, order granting summary judgment to Owens-Corning. As of the date Owens-Corning filed its summary judgment motion, Shine had failed to identify any Owens-Corning products to which her husband was exposed, or any witnesses establishing that exposure. Under Illinois law, which governs this diversity case, Shine must identify the allegedly harmful product and demonstrate a link between that product and each defendant. Schmidt v. Archer Iron Works, Inc., 44 Ill.2d 401, 256 N.E.2d 6, 8, cert. denied, 398 U.S. 959, 90 S.Ct. 2173, 26 L.Ed.2d 544 (1970). To satisfy her burden, Shine was required to specifically identify the asbestos product to which her husband allegedly was exposed, Tragarz v. Keene Corp., 734 F.Supp. 834, 836 (N.D.Ill.1990); Leng v. Celotex Corp., 196 Ill.App.3d 647, 143 Ill.Dec. 533, 554 N.E.2d 468 (1990), and show that he worked close to the product at the time it was being used. Tragarz, 734 F.Supp. at 836; Wehmeier v. UNR Indus., Inc., 213 Ill.App.3d 6, 157 Ill.Dec. 251, 266, 572 N.E.2d 320, 335 (1991) (plaintiff must prove “exposure to a specific product on a regular basis over some extended period of time in proximity to where the plaintiff actually worked.”); Zimmer v. Celotex Corp., 192 Ill.App.3d 1088, 140 Ill.Dec. 230, 233, 549 N.E.2d 881, 884 (1989); Naden v. Celotex Corp., 190 Ill.App.3d 410, 137 Ill.Dec. 821, 824, 546 N.E.2d 766, 769 (1989). These things she did not do. Given the exclusion of her product identification witnesses, it is clear that the district court was correct in determining that no genuine issue of material fact existed. The grant of summary judgment to Owens-Corning was proper.
Shine also disputes the June 25, 1991, grant of summary judgment to Owens-Illinois and Keene. In responding to this summary judgment motion, Shine submitted deposition testimony of her husband. Owens-Illinois and Keene maintain that the deposition, which had been obtained for use in separate litigation involving a different plaintiff, is inadmissible under Fed.R.Evid. 804(b)(1), and that in any case it failed to raise a material issue regarding causation. We need not reach the issue of admissibility, for even if the deposition is considered, we agree with the district court that it is insufficient to establish a genuine issue of material fact.
Although in his deposition Mr. Shine stated that he used “Kaylo,” an asbestos-containing product manufactured by Owens-Illinois, a review of his testimony indicates that he used that term generically, rather than as a brand name. Moreover, when asked specifically about Owens-Illinois at the job site, the most Mr. Shine was able to offer was a recollection that he had seen boxes of Owens-Illinois material present at the job site — which under Illinois law is not sufficient to create a material issue of fact regarding causation. Although Mr. Shine stated that Owens-Illinois products were present and used at one of his job sites, as the district court recognized, he did not testify that he worked with or was exposed to Owens-Illinois products. Nor did he testify that he worked in close proximity to co-workers who were exposed to Owens-Illinois products. Without direct evidence of exposure, the plaintiff must establish an inference of probability, rather than a mere possibility, of exposure. Tragarz, 734 F.Supp. at 836; Wehmeier, 157 Ill.Dec. at 267, 572 N.E.2d at 336. We agree with the district court that Shine failed to do so here, and find that the grant of summary judgment to Owens-Illinois was proper.
With regard to Keene, whose predecessor’s former subsidiary, Baldwin-Ehret-Hill, manufactured asbestos-containing products, Mr. Shine’s deposition testimony was tentative at best. Although he stated *98that Keene or Baldwin-Ehret-Hill products were present at a past job site, he could recall no specific products. Even assuming that such products were indeed present, there was no evidence to establish that Mr. Shine had worked with or was exposed to the products. The most that can be said is that Keene products were at a job site where Mr. Shine worked. But again, the mere presence of Keene products at the job site is insufficient to create an issue of material fact. Tragarz, 734 F.Supp. at 837. Given the lack of direct evidence of exposure, as well as the absence of circumstantial evidence, the district court again acted appropriately in granting summary judgment to Keene.
Although Shine also conténds that the district court erred in denying her motion to reconsider the order granting summary judgment in favor of Owens-Corning, we find no merit in this contention.
We sympathize with Mrs. Shine, for the exclusion of the product identification witnesses had the practical effect of dismissing her case. However, it is the district court’s prerogative to adhere to established schedules and to exclude evidence provided in derogation of the timelines established therein. In light of the recent cases indicating the district courts’ willingness to strictly adhere to court-ordered deadlines, see Siwy, 1991 WL 139996, 1991 U.S.Dist. Lexis 10074; Fitzgerald, 1991 WL 136019, 1991 U.S.Dist. Lexis 9713; Trzesniak, 1991 WL 60521, 1991 U.S.Dist. Lexis 4636, we suggest that counsel pay particular heed to time frames, and discuss any potential ambiguities or difficulties in meeting them with the district judge assigned to their case, lest their clients discover all too late the penalties for noncompliance.
Affirmed.