FaceTime: server failed to process registration data

I can't recall having the exact same issue with FaceTime because I didn't use it but yesterday I was unable to start using it: I entered correct my Apple ID credentials. Then, on the next prompt me entering the second e-mail address (in the address field, not in another AppleID account field!), one meant to be visible to my contacts (just one I added to my mail addresses list on Apple ID manager pager) and hitting "Continue" it displayed "Verified" briefly, then threw me back to FaceTime...

FaceTime: server failed to process registration data