Well, technically, there doesn't HAVE to be a limit. But some states feel that some income is better than no income.

As I said, my state is not Illinois, but the tendency is that if you turn down work, they're almost sure to turn you down. However, if you accept part time or lower paid work, you can often (and I've had this happen myself when I was laid off) be paid at least a partial unemployment benefit to make up some of the difference.

I was laid off from my HR Director position when my company was on the verge of bankrupcy. I was granted unemployment; I took a six week temporary job in the middle of my claim and was able to pick up the remainder of my claim when it was over. This was just after the dot-com crash and there were 300 applications being submitted for each individual position; by the time my UI claim was ending I had so far always been among the 299 unsuccessful candidates. So to make sure I had at least some income I accepted an offer of help from a friend and took a position cleaning hotel rooms at the hotel she managed. The UI office decided that since I was clearly making every effort to find work and not being picky about it they would grant me a second, partial claim even though I was working, since my new job was only a quarter of my previous pay. I was given another 26 weeks of partial benefits. But if I had turned down the temporary job because I only wanted full time, and turned down the housekeeper job because I only wanted work in my field, they wouldn't have given me a penny once my initial claim ran out. Do you see where I'm going with this? She could have accepted the part time work so that she'd have at least some income, probably been granted partial UI, and continued looking for full time work. Instead she turned down the only offer on the table at that time.

Not all states take the same position and as I said before, she is legally entitled to appeal the decision. But while I've never had employees in Illinois so I can't say for certain, I've heard through the HR grapevine that IL may be taking the same position as my state. So I can't be overly optimistic.