HI,
In these cases Apple tends to be aware at a very high engineering level because some Level 2 support people have been very savvy about the information they have gained and the way they have passed it on.
On the other hand though those engineers don't have a fix so no support Document is issued, the staff at Apple Support or indeed Stores have not been told - so it can look like no-one knows what is going on.
These are the 4 original error messages Get help signing in to FaceTime, Game Center, Messages - Apple Support
It have recently been rewritten to include 2 step verification.
This is a pic of another Error Message that I call number 5

It is very clear that the app has not logged in the iMessage account and to contact Apple Support.
There then seems to be two silent failures to login properly or fully.
The 1st of these (or Number 6) involves all Contact showing Red as you Send an iMessage (Not Registered)
The 2nd (Number 7) is when you have Sent but the info comes back "Not Delivered" for each iMessages.
Both these are in spite of the Apple ID working on other devices.
Each Login by each device is unique and that some info about the device is used to create the specific Registration.
In the case of a Mac and since the OS X 10.8.2 update this has been the Serial Number of the Mac which has been used to create an "Auth Token"
Some have reported that deleting everything that mentions all or part of IDS: AppleID@someServer.com-Auth Token should be deleted from the Keychain (Applications/Utilities)
The app should be Quit when you do this.
If this does not fix it then the ONLY fix will be to contact Apple Support and speak to a Level 2 person and ask them to check the Registration of your Mac on the iMessages servers.
This is an Apple ID issue and not an Messages/iMessages account issue (Software) or a Hardware issue.
Without the error 5 Messages though it can be difficult to persuade them to check without jumping through some hoops.

8:09 pm Thursday; March 12, 2015
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