The short answer is no, the long answer is maybe, but it depends on the cause of action, choice of law clauses in certain contracts, special rules for medical and legal malpractice etc.

Each state actually has its own laws for limits on different types of damages. (This has in the past, created problems such as forum shopping jurisdictions to look for the most favorable laws, if one could maintain that jurisdiction is proper in that forum). Texas was an example for a while of a state that had some problems with that. Additionally, choice of law clauses in certain contracts can create a quagmire here as law (as even though the forum might be one state, the applied contract law could be another states due to a choice of law clause). Punitive damages can be covered by insurance, but first, they are rather difficult to get, and do not exist for every type of statutory violation or cause of action and second, often if some kind of behavior or action) results in punitive damages, it may violate that parties particular insurance policy. (which sounds similar to what you are explaining)

There are a few examples of federal cases that address this, but looking to your IL laws that could point you in the right directions. Here is one example. http://www.ilga.gov/legislation/ilcs/fulltext.asp?DocName=073500050K2-1115.05

I also don't know what the hypothetical cause of action is, but it can very depending on whether or not it is a personal injury/work injury case vs. a breach of contract case or other statutory violation matter. Check out these Illinois pattern Jury instructions (although a quick search on west-law might do you better) http://www.illinoiscourts.gov/CircuitCourt/CivilJuryInstructions/35.00.pdf