Yes it can happen. Of course it can happen. It's rare, but it has been reported on the forum before, usually (in my limited reading experience) by a Bagle worm variant.
To be fair to Avast, there is malware around - lots of it- who's first mission is to disable security programs,(for obvious reasons) and it happens quite a bit, and to other AV's also.
I think it happens a lot less with Avast now that there is a self protection module, but basically, if you don't learn to take the steps to harden the browser, (includes immunizing, and limiting the rights of any old application that might be running in it) and keeping software up to date, this could happen. Chances are probably small, but it could happen.
The trick is to close as many of the vulnerabilities that a Windows based computer typically has to prevent the malware downloading and installing in the first place.