The only cause to this problem, or pattern to its occurrence, I can come up with is a loose theory that it happens during periods of inactivity. My hunch is that when you appear "idle" AIM somehow misinterprets this as a disconnect. Locally you remain logged in, but AIM thinks otherwise. Luckily I may have discovered a workaround.
Launch iChat (quitting Adium is not required) and go into Preferences. Click on Accounts. Make sure your AIM account is selected, then click on the Security tab. Changing any of the settings in this options panel, then returning them to their original state, somehow magically vexes the problem, at least so far as I can tell.
I don't know whom to point fingers at in this case; AIM or Adium. Both begin with the letter A, so that makes each of them equally culpable in my opinion. Wherever the guilt may lie, I just want the problem fixed ASAP.