Hi,
The Original Poster said it was with a New computer in which you get Apple Care.
However we know that the Apple ID used by the Mac for iMessages can randomly get blocked at the iMessages server.
It's supposed to be a security feature but some people seem to find it gets triggered without any apparent input in Messages.
It does seem to centre around the OS X 10.11 Upgrade and then the Updates at 10.11.3 and 10.11.5 (Yosemite also had some but less issues and it happened before that as well).
The Only Fix is to speak to Apple Care (Apple Support) and say you have an Apple ID issue (Free to fix if your Apple Care has officially finished).
It then takes time to get elevated past the level 1 responders who are following scripts. (Be polite and insist on speaking to someone higher).
Some Level 2s know about the issue but do seem to be tasked with trying things you may not have and data gathering.
Reinstalling, Upgrading or Updating do not work.
Eventually you get told the Engineers will have to reset it and that can be between 24 hours and 5 days judging by the posts others have made when they do report back.
Some specifically note the follow-up phone call is often missing.
8:17 pm Wednesday; August 3, 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)