The situation is different at almost every dealership. The large multi branded
dealers have set standards for this. I have found the normal amount allowed to be between 1000 and 5000 dollars.
That being said I have seen a dealer take the whole amount on the credit card
for a car they either wanted to move or they had enough profit built in to cover their fees.
The bottom line is when you have come to the final moment of the negotiation and you are about to say yes to the price, mention that you will only accept the deal if they will accept XXX dollars as payment.(or down payment).
By this time they have spent a lot of time with you and do not want to lose a deal. They will probably give you some bs at that point, but if you stick to your guns, they will, at some point, accept more cc monies.

good luck