I know this is an old thread, but I am having the same problem. I can add all but one card to Apple Pay. The card that will not work is a Capital One Visa card. The problem occurred for me after I restored my phone from a backup—I had to add all of my cards back to Apple Pay.
I have done every troubleshooting step I can think of, and I have been advised by Capital One and Apple phone support. Capital One is "escalating the issue" and I have to wait 7-10 business days for an answer (usually do not have luck with escalations), so I thought I'd see if anyone had any other suggestions.
The problem is only with this one card. It worked with Apple Pay fine before the restore and now I get the error message "Can Not Add Card. Try again later or contact your card issuer for more information."
The more advanced troubleshooting steps I have taken are as follows:
Erased all content and settings, restore from iCloud backup
Erased all content and settings, restore from oldest iCloud backup
Erased all content and settings, restore from iTunes backup
Erased all content and settings and tried adding the card before restoring from a backup
Put my phone in to DFU mode and then reinstalled iOS 10 from iTunes, then restored from iTunes backup. I was trying to downgrade the iOS back to 9.3, but I was only given the option to download to iOS 10.
Tried adding my card to a different iPhone running iOS 10 and also iOS 9.3. I can add my other cards to these devices, but not this Capital One card.
Any other suggestions would be much appreciated!