There may be something in the Console App (Applications/Utilities) although iChat and Messages tend to be relatively good at at least telling you something.
This sounds like it is not even getting a response from the Google server.
Check the iChat Menu > Preferences > Accounts > Google Account.
In Server Settings does it list Port 5223, talk.google.com and is using SSL ?
Go to the Account Info tab and untick "Use this Account"
Go back to Server Settings and Change the port to 443 (Google accepts logins on this port)
You may also want to check your router is allowing connections on port 5223.
(If it is using Port Forwarding to do this is the computer at the IP that the Port Forwarding is pointing to)
If you are using UPnP then there should be no problem.
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
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
"Limit the Logs to the Bits above Binary Images." No, Seriously