1) As the two linked threads show the most common issue for an AIM login is the SSL server at the AIM end.
Since it was included in iChat there has been an issue with reconnects if the service is dropped.
The NON SSL connections will reconnect but the SSL ones need it to be manually reset. (essentially setting the Mac end to Log out then Log in again.)
2) Apart from this there can be issues if the connection was cut and then the Mac Shut down.
This leaves the com.apple.iChat.AIM.plist (still used with Messages) as listing one set of info about the State of the Log in and the AIM server having another.
If a Log out, Log in does not work in 1) then delete the listed .plist
In the Finder > Go menu > Go to Folder and type ~/Library/Preferences
Find the .plist and drag it to the Trash
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