A reboot fixed it. Crap! I could have gotten another 1 1/2 h of sleep last night, instead of barking up all sorts of wrong trees.
Now I have to see, what might have caused this and whether it'll be working at home. I did quickly sleep cycle (put to sleep/wake up) my MacBook Pro, in order to see, whether that was what triggered it. However, it wasn't.
At various points during it's start Messages will write bits of Info to the relevant .plists involved.
One of these bits of info is the Internet Speed it can "see".
This can be effected if the app is starting up at the same time as the computer.
It can also be effected if the computer is busy with something like a download or Video Streaming.
Once the info is Written it does pay to try a restart of the App and then the computer if the tilde (Disconnected) icon appears.
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