Hi,
You mean this error message
It only happens in Mavericks in relation to to the OS X 10.8.2 update to Mavericks.
This updates the Messages app to version 7.0.1 where it starts to use the Mac's Serial Number as part of the Login process.
First of all check the Serial Number can be seen.
Go to the Apple Icon Menu top left and select About this Mac.
In the panel that opens click twice on the line that shows you the Version number of the OS.
It will change to the Build Number on the first Click then show you the Serial Number
til you get to 
If the number is missing then you will need to go to an Apple Store and have it put back.
It sometimes gets missed if the Logic/Mother board has been replaced during a Repair or Refurbishment.
Once that is done you can contact Apple Support (Same as Apple Care).
You have an Apple ID issue and you should tell them this. (free to Fix)
It is unlikely any current Level one person will remember this as Mavericks was originally released some time ago.
Insist on speaking to someone higher if necessary.
They should be able to look at the iMessages Servers and check if your Mac's use of the Apple ID has been blocked.
If you Mac could not provide this then the security feature that this Update brought is working properly.
At the time this originally came out Level 2 people were resetting the Server with no problems.
Now similar issues are showing up all the way to El Capitan - not so much about the missing Serial Number but issues with the feature being triggered when it shouldn't.
This mean it can mean you have to get all the way to the Engineers before they reset the server, and that can still take "overnight" or 24 hours.
Updating further within Mavericks will not solve this.
Neither will Upgrading to a later OS X version as the Server will still need to be reset after the Serial Number has been sorted.

9:26 pm Monday; May 16, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone and an iPad (2)