Most credit cards let you set the spending limit as low as you like (I don't know if there's a floor, but I would bet you could set it to $100. )

I could have sworn credit cards will be rejected if you hit your credit limit.

If you try to get some "unreasonable" amount of money from them - probably. Anything that you're remotely likely to pay off in penalty charges is unlikely to get blocked. That means an unauthorized overdraft of $1k that you don't have is a good deal for the bank.

Hmm. I guess I don't really know for sure, my credit limits have always been much, much higher than my spending.