I upgraded to 10.8.2 today and cannot login to Messages, Facetime or Game Center. In Messages I get the error "Could not sign in to iMessage. The server encountered an error processing registration. Please try again later. " when I enter my Apple ID. It was working this morning prior to the update. In FaceTime i get the error "The server encountered an error processing registration. Please try again later.". I was able to also use this in the prior version (10.8.1).
Game Center on the other hand; I've never been able to get that to login. I immediately get an error when I try to login: "The requested operation could not be completed due to an error communicatiing with the server." I was hoping today's update would fix this. (I don't use it often. And it is a different Apple ID than I use for messages/facetime.
It looks like an problem on Apple's account side, but I was able to log in to the support community with my Apple ID, and my iPhone/iPad. Hopefully, I'm not the only one, and we can get this fixed.
Check that your hosts file (/etc/host) is not blocking some important communication with Messages or game center.
I opened my hosts file in TextMate and commented out everything that was not default, saved it, restarted Messages and GameCenter, then logged in.
After logging in I uncommented the comments in my hosts file, saved it, and everything was back to normal.
Hope this helps someone.