Apple made an agreement with AIM (AOL Instant Messenger) when the first version of iChat canme out that Apple IDs issued by Apple would be also Valid AIM Screen Names.
At that time that was the @mac.com names issued under the .Mac Service. These are still valid even if you had a trial version.
Later it was @me.com names issued under MobileMe and more recently those issued under iCloud along with the new @icloud IDs that have also been launched.
IF you had a paid for @me.com name when MobileMe started up you could link the names for email purposes. (effectively it is one accounts with two names)
The same thing has happen now with @icloud.com ID s being also given out if you registered an @me.com name under iCloud.
However these individual names are separate AIM Screen Names when you go to log in via iChat.
There are a few issues depending on which version of iChat and which ID registration process you have gone through.
1) All these IDs (@mac.om, @me.com and @icloud.com) all have to have a password of 16 characters of less due to the limit of the AIM servers.
2) With iCloud Issued @me.com and @icloud.com names that are used in iChat 6 and Mesages in Mountain Lion, Apple have the app logging in to me.com as well to then allow the AIM servers to "see" the password.
This means iCloud issued IDs will not work with iChat 5 or earlier versions.
The most likely cause is that the password you have for your Apple ID is too long to work with the AIM servers.
It is possible that you have an ID that will not work with your version of iChat despite being a valid ID.
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