I bought a Hampton Bay ceiling fan about two months ago. We typically leave it on the "low" fan setting. Today, it suddenly started changing fan speed to medium or high 30 seconds after we set it on the low setting.

Any idea what could make a fan speed up that way?

Is it a remote fan?

Yes, but the remotes aren't the issue. I took the batteries out, and they still changed. The most impressive one was that the fan also started to switch ON from an off position. And no, my house is not haunted... :D

Switch the frequency on the fan and remote, right now they're on the same frequency as something else in the house.

http://www.ceiling-fans-n-more.com/ceiling-fan-remote-controls.php

I'll give it a shot. Is that something that would have started with no warning in the last 24 hours? The fan is 3 months old.

Sure if you or someone in your neighborhood bought and started using a new RF device in the last 24 hours