Seems to mostly happen to me when TB checks my hotmail account in the background.

I used to occasionally get "STAT Command failed" errors, something about it was only allowed every 15 minutes which must be a hotmail pop3 policy - maybe this is that same problem just popping up in a new form under 7.01

Have been receiving above error for days now. Tried Paulbrain suggestion and changed msn email account server settings to check mail ea 15 min (was set for 10 min) and have not received the error message again. Because I was receiving this message frequently (possibly ea 10 min when TB was checking msn email) and have not received the error message since changing server settings (2 hours ago) I am inclined to say that this solved my problem.

UPDATE: Had TB minimized for over a day and did not receive the error UNTIL I opened my live messenger chat application in conjunction to having TB already opened. Closed live chat application and have not gotten the error for over a day now. Inclined to believe that there is a connection.

Mmm, interesting. Not been online as much this week, but the 'security library failure' message has stopped (I changed nothing), but now when in Thunderbird asking to get mail, I get a message in the bottom left of the screen stating 'Looked up pop3.live.com...'. And nothing happens. Not helpful. And this is at the start of the day when no mail has been received for two days, so it does not appear to be linked to any 10-15 minute delay linked to picking up mail from my Live account. Any ideas? I used to think Thunderbird was very good, but it has been a pain this week!

Thunderbird support is over here.
http://www.mozillamessaging.com/en-US/support/
or here:
http://forums.mozillazine.org/viewforum.php?f=39