Hi,
I have exactly the same problem. I sign-on correctly, and that is accepted, then I try to proceed with Facetime startup and fails, and returns to authentication, with "The server encountered an error processing registration. Please try again later."
I have replaced the motherboard in my late 2006 IMac, and it has no serial number, never had one for the NEW motherboard. Facetime is the only thing that seems to kick a fit because of it. My IPhone can Facetime just fine, using my same ID. No problems, so it's not my ID, and it is not a public Apple Server problem. Has to be something like a computer hardware ID problem (something that should be anonymous).
So while everything is fine in Apple land, they made the authentication system too picky, for data that should be anonymous. Since Facetime cannot be installed on something other than a Apple product, you have to wonder what they were thinking. I have been a non-certified Apple tech in my community for more than 10-years. I've never seen anything like this. It is annoying. What's worse, for those rare cases where people manage to build a FrankenMac, since there is a substitute for Facetime, incommon usage (namely Skype); where's the big whoop about the serial number. Sheesh!