Okay, I'll take it back. It looks like I'm logged out/logged in depending on which page I'm on. For instance, I'm logged out on the main page but logged in when I visit a thread. The problem is, that I'm getting the same cookies for different paths (and with different values) and only one of those is actually valid.
e.g.
ubbt_mysess with path = "/" and "/ubb/"
ubbt_myid with path = "/" and "/ubb/ubbthreads.php/ubb/"
There's also several login and session cookies around and I have no idea if they're all valid/needed/contradictory.
I'm not familiar with the UBB forum software, so no idea which cookies are correct/duplicates/config-issues, etc, but it certainly feels like something's broken there.
Next thing I'll try is delete all cookies and start from scratch.

Michael