Hummm,
I have feeling that uptake of Mountain Lion is happening in waves.
Add to that the increase in Messages due to the Olympics and I think we may have an issue with the Servers Apple use for iMessages.
Of course such a statement is of no use in finding a solution and is only guess work as to a possible cause.
I would be tempted to try a "Repair Permissions" in Disk Utility > First Aid tab
and I would also try a PRAM reset.
Shut down the computer.
Restart it holding down Apple/CMD ( or ⌘)+ALT( ⌥)+P+R Keys until you have heard three Start Up Bongs.
9:28 PM Wednesday; August 15, 2012
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Lion 10.7.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
"Limit the Logs to the Bits above Binary Images." No, Seriously