When filing Requests for Admissions, send them to opposing counsel by certified mail. It's the only way to deal with the first argument you'll hear when you try to get the requests deemed admitted for a failure to respond: "But I never received your Requests for Admissions!"
Even this may not be a good excuse, however. In Glasco v. Marony, the Fifth District discussed what constitutes good cause for allowing an attorney to respond to Requests for Admissions out of time:
We find that the circuit court did not abuse its discretion by denying the plaintiff's request for an extension of time to respond to the defendants' request for admissions. Mistake, inadvertence, or simple attorney neglect cannot constitute the sole basis for a good cause determination. See Greene v. City of Chicago, 48 Ill. App. 3d 502, 513 (1976), aff'd, 73 Ill. 2d 100, 382 N.E.2d 1205 (1978); Floyd v. United States, 900 F.2d 1045, 1048 (7th Cir. 1990). We affirm the circuit court's judgment on this issue.
In Glasco, the court rejected the responding attorney's excuse that "he had been out of town and without a secretary at the time of service," and so should be excused for filing his responses late.