Sometimes when I use my computer, the cursor starts moving on it's own and clicks on things.

It seems the following story is not relevant to your case, mlb, but it might give a heads-up to others.

A colleague noticed this happening on his computer and it had the institution's network techo's hopping for a minute or two. It turned out he was using a wireless mouse and keyboard, as was another colleague in an adjacent office. One was on a Mac and the other a Dell using XPII and they were using quite different applications at the time. It turned out both were using wireless keyboards & mice and both were on the same frequency, which managed to penetrate a good 40cm (15") of reinforced concrete wall as well as several metres of airspace.   

A trap for the unwary.

Cheers, Rowan