Help! My palm tree is sick!

Just looking at it, the plant looks green and healthy, and new fronds are sprouting and opening up all the time. My oldest daughter bought this for me for Father's Day in 2013, so it means a lot to me.

This evening I noticed a couple of branches in the back that were brown and dying. Upon further inspection, I noticed spots on them (photo's below). Is it some kind of mite? Are my other houseplants in danger?

I searched around a little bit, but didn't find much help online. Does anyone know what I need to do to save this palm? It would break my heart to lose it!