Hi,
Unfortunately we are seeing more and more of this.
Each device you have registered with iMessages has it's own Login with a unique identifier.
For an Mac this tends to be the Auth Token for IDS shown in the Keychain which uses the Password and the Mac's Serial number.
Open Keychain Access in Applications/Utilities.
Select Login top left and Passwords lower left.
On the right you should find "IDS: yourAppleIDasEMail-Auth Token" where yourAppleIDasEMail is an email ID you use in Messages for iMessages.
With the Messages app shut down try removing this item then Restarting the app.
It should be recreated if you are connected to the Servers (it still might not get a full login though)
if you get a partial result then go to ~/Library/Preferences (Hold the ALT key whilst using the Finder's Go Menu and select the Library).
Find com.apple.imessages.bag.plist and delete it (Drag to Trash).
Restart Messages as this one holds the server name and will cause the iMessages account to recreated it potentially with new info.
Do the same for com.apple.ids.service.com.apple.madrid.plist (It is more drastic)
Restarting Messages will need you to re-enter the Apple ID as this .plist holds the Apple ID and the Alternatives meaning the pop ups about other devices will also app and need to be accepted.
Check System Preferences > Network > Advanced Button > DNS tab.
This should list (either in greyed out form from the router) or an actual DNS Server IP address (or two). (Preferably not the router's own IP).
In some cases the issue is with the server end.
We know from other errors that you have to contact Apple Support in some cases.
You will need to ask to speak to a Level 2 person (Level ones use Scripts and tend to look for Software and Hardware issue and whether you have Apple Care).
Account issues (which is what this is) are free to fix.
Some Level 2s will have heard of this and convincing them to check your Mac's use of the Apple ID at the iMessages servers will not be too difficult.
In other cases they will want your to jump though some more hoops and in some cases they will also be talking to the Engineers looking for an Answer.
If th last happens you may have to wait overnight for a reset.
9:08 pm Friday; May 22, 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
