It would help to know what version of the OS X the "new" computer is using as this will tell us which version of iChat or Messages you have.
"Old Username" sounds like it should be an AIM name or one that is valid with AIM.
Some Apple IDs are valid as AIM Screen names (Those ending in @mac.com, @me.com and @icloud.com)
You may have changed the password on this account and it may now be longer than the 16 character max that the AIM servers work with.
If you had an @me.com name and "converted" this to an iCloud @me.com ID then this will not work in iChat 5 or Earlier.
This is because iCloud IDs used in iChat 6 and Messages in Mountain Lion require a second Login to Apple that then allows AIM to "See" the password.
iChat 5 and earlier versions do not have this additional security check.
If it is an @mac.com name or a plain AIM name then try logging in here
This is an additional Settings page at AIM which may tell if the account is suspended (in the past AIM have re-released Names they felt were Unused back in to public use).
More recently they have "suspended" some accounts when people have used them in different countries when travelling.
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.4)
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