I have a suspicion about what the problem is for me. SE will not confirm or deny.
My credit card name doesn't match the name on my account, because I created my account with a nickname.
SE won't let me change the name on my account. And I'm not very well going to get a new credit card with my nickname. Therefore I'm hosed.
My wife's card has the same CC# as mine. Registering it on her account worked fine. SE just feigns ignorance and claims "It's the bank!" when it obviously is not.