HI,
It depends on a couple of settings one of which is in Messages and one ON line at the AIM site.
It also depends on what Screen Name you are using.
First Messages.
The Line indicated by the Red Arrows needs to be Unticked to Receive Off Line IMs from AIM (and other services that may do this)
2) the On Line spot at AIM Settings
Log In.
Check Off Line IM menu item (As you may have set it in AIM on a PC or a On line in the past or in response to the first Off Line IM you got and replying No).
In the past in version of iChat the AIM Settings page was the "better" palce to do this and was more certain of "Blocking" Off Line IMs
Since about iChat 5 the iChat setting has improved and is more "complete" in the way it blocks (or not) all services.
In iChat 5, 6 and Messages you could rely on the Preferences Setting to stop Off Line IMs
The AIM settings page can be used for AIM registered Names and Apple @Mac.com names.
Apple @me.com names from iCloud Registration are valid AIM names but iChat 6 and Messages make a second login to me.com that then allows the AIM servers to "See" the password.
The AIM site does not have this Ability and you cannot login in the AIM Site (The names also don't work in iChat 5 or Earlier)
8:36 PM Saturday; September 1, 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