Hi,
Despite Apple owing the Site and providing Hosts (Moderator) staff and some "Community Specialists" they rarely read the Threads for issues.
Sometimes some issues will get noticed and an Apple Doc set up with what is thought to be the fix (in most cases that was also discovered on the boards by trial and error) such as this one FaceTime, Game Center, Messages: Troubleshooting sign in issues
In the case of this error the Regular Posters have been running through a gambit of ideas with people who have posted about the problem.
After one person did a Clean Install (Erased the Drive and created a new Mac User Account without migrating from a Backup) and set up the iMessages account in the Messages app that way it become clear the issue is either the Mac Serial Number which is used for an Auth Token in the Keychain or something about the Account on the iMessages server.
Based on that info I have Sent Feedback (and suggest other people do as well)
And I posted in the Lounge (where Level 6 Posters and Above can post) and reported the problem in that hope that two routes are better than one.
It really doesn't matter how much you love Pete, pleading here does very little.

8:02 pm Friday; October 3, 2014
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad