Hi,
The longest I have seen posted here for it to work is 11 days after the Device last made contact with the Server.
The Server keeps records of the devices Registered for iMessages.
It effectively "Copies" iMessages so they can be delivered individually to each device.
The Servers keep pushing the iMessages until they are accepted - although there does appear to be a Time Limit to this.
Once the iMessages have been delivered then the servers delete them.
It appears if the Time Limit is reached they are also deleted.
As far as I can tell the App is working as it should.
7:38 pm Saturday; February 6, 2016
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