That's an interesting question. The first digit specifies the card type. 3 is AmEx, 4 is Visa, 5 is Mastercard, 6 is Discover. I'm sure there are some other lesser-known ones. After that, there's effectively a single checksum digit. So that's a 14-decimal-digit address space for Visa, Mastercard, and Discover, which is 100 quadrillion combinations. (13-digit for AmEx, so only 10 quadrillion.) So that's like 50,000 numbers for each person on Earth. Probably not a big deal.

I've glossed over a lot of details that actually make the numbers less today than what I've quoted, but the fact of the matter is that there's a single checksum digit, so assuming that the cards expanded to fill the 16-digit address space, that's one quintillion numbers.