I was also having the "No Steam Logon" issue a few days ago... join, kicked... join, kicked... it is rather annoying... I looked it up on Steams site, and they had some lame message about restarting steam, and if it continues, just be patient as the servers are probably busy...
yeah... bs. Steam just isn't quite there yet for stability.
EDIT: oh yeah, I just kept rebooting steam, and eventually it worked.