Hi,
I am following the whole iChat Community - meaning I get an email fro every post that is made there, which is where this Thread started off.
You Original Post does not say if it is the iPad or the Mac that is having problems.
iMessages have there own Servers.
On the Mac in System Preferences > iCloud there is nothing for Messages (or iMessages) iTunes, FaceTime, Game Centre, iBooks or several other non iCloud login places).
The iMessages servers check the Apple ID site for the password details.
Also the iPhone and iPads using iOS use the Serial Number as well as the Login info to create a device specific login.
This also happens on the Mac version.
The combo of the Apple ID , Password and Serial Number generates an Auth Token for IDS in the Keychain.
In Applications/utilities open KeychainAccess.
Set "Login" top left.
Set "Passwords" lower left.
On the right find the item that reads "IDS: yourAppleID- Auth Token" where the italics bit is your Apple ID email.
In the Apple Icon Menu top left use the Menu to open "About this Mac".
In the Mavericks version click the line that tells you the Mac OS X version
This will be the Build Number (another way Apple Counts the versions)
Click it again and it will reveal the Serial Number.
Don't tell us but is it present ?
If it is try these two things (test Messages again in between)
How to Reset NVRAM on your Mac - Apple Support
Safe Boot
If it does not work - and we are talking about the Mac - then you will have to contact Apple Support (Apple Care)
Level one responders are Script led and trying to solve Software and Hardware issues under Apple Care.
Tell them you have an Apple ID issue.
As the Devices can be blocked individually at the server your Apple ID in iMessages has become blocked on the Server.
Apple are aware that this sometime happens but not why it happens.
You are likely to need to ask to speak to someone higher.
They still may want to do some further test and data gathering.
You WILL NOT need to Update to the last version of Mavericks if you are not using it.
You WILL NOT need to Upgrade to the next or later OS version.
You WILL NOT need to Reinstall as the issue is at the server end.
Ask to be bumped higher if you are not getting anywhere.
Ask the person to check your ID for it's use by you Mac on the server.
It might take the engineers to be involved and sometime you get told it will take 24 hours to reset.
We have not found a quicker method or set of word that gets this sorted any faster.

7:48 pm Tuesday; May 10, 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)