The solution to this is three words: We have neighbors....
We live in a townhouse...our neighbors probably have the same fan/light combo with the same remote. When they point it towards us, they turn our light and fan on as well as theirs.
Oddly simple solution is a really weird problem. Now I just have to work up the guts to talk to our neighbors. The funny thing is when they kept turning our fan on one night, my husband turned the remote towards their wall and flickered it a couple of times. I think they got the message since we haven't really had a problem since then.
Did any one guess correctly?