A quick search shows that some great deal of people are having this problem. And it included me, until now.
After being plagued by this very problem this evening (around 10 disconnects in a row), I decided to end this menace. Especially since I often had the same problem in the past.
So I was searching for a solution on the Internet, skipping very old or trivial ones (such as reinstalling MSN or deleting some roaming appdata folder of MSN Messenger), when I stumbled on a solution that worked for me.
So if you’re reading this and are having the same problem AND using AVG Antivirus, you can stop MSN Messenger from signing you out by simply disabling AVG’s “Instant Messaging protection” (located in Overview->Online Shield->Instant Messaging OR Tools->Advanced Settings->Online Shield->Instant Messaging). Optionally you can only disable MSN Protection.