Hi,
There are several different type of error messages.
There are the four supposedly covered by this Doc FaceTime, Game Center, Messages: Troubleshooting sign in issues
There is a fifth where the User gets a longer Error message with instruction to Contact Apple Support and give them an Error code number.
this tends to be free but it does help if you explain it is an Account issue (that involves iMessages in the Messages App on the Mac).
Ask to speak to a Level 2 Support person if you get baulked.
More recently (for many the 9th September seems to be the key) there have been many more who don't see and error messages but do get told none of their Contacts are "registered with iMessage" which is an error as they work on iPhone and iPads. (Although iPads can be in either group)
Today I posted in the Apple Support Communities Lounge (Level 6 posters and above) where the Hosts will hopefully push the info up the line. (it was posted with the "New Technical Issues" tag).
I also posted feedback to the iChat feedback page. I use this ones as Messages came out of iChat.
The issue with this is the time it takes to cycle through the process.
Only twice have apple provided an Standalone Update to iChat or Messages and the last of those was at iChat 3.
There may be a 6th error message that tell you "the Device does not have the appropriate Credentials".
I have only seen one post listing this and the issue is not resolved (the Original Poster has yet to return).

9:45 pm Tuesday; September 23, 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