The probably do, but (I think) OP is asking about using the card directly for a large purchase which would not incur the fee.
I had friends do this with guaranteed student loans in the 80's. Took out the loans which accrue no interest while enrolled full time, put the money in CD's, and collected the interest while they had the 0% rates on the loans. I'm assuming that they did the math on the loan origination fees, etc. to make sure it was still a good deal, sort of like you'd need to do if the CC has an annual fee that you'd have to take into account in the math. It works as long as you are very disciplined. The biggest reason not to do this is that you might make a mistake and then get penalized by the CC company. It's basically dollar lending arbitrage that will be there hanging over your head the whole time. If that doesn't stress you out, then go for it.