Mac OS X Messages and Facetime Started Failing
The best answer so far (weeks after the problem) was at https://discussions.apple.com/message/27149499#27149499 The gist of it is a corrupt keychain. I'll copy paste the steps for a fix here but I advise to read the actual thread over at Apple Discussions and have backups.

Here are the steps:


Open Key Chain Access
Select Login at the top left side
Select All Items on the lower left side
Search for entries of IDS.
Highlight and delete all entries that has IDS. (For me I am going to
need to re-install my RSA SecurID Software Token)
Search for entries of FaceTime
Highlight and delete all entries of FaceTime
Search for entries of Messages
Highlight and delete all entries of Messages


I hope this helps.