Can You Catch A Cold From Your Pets?

Over the weekend, I noticed our cat Oswald sneezing quite a bit.  I'm not sure what's going on with him, but I think I caught his cold!

I told my husband we're going to have to take him to the vet. Meantime, I held him and tried to make him as comfortable as possible.  Now today I woke up with a really bad cold, did I catch it from my cat?

Is it possible that when your pets are sick, you can become sick from them?  I really didn't think that we could exchange germs, but I found it to be a bit odd that I all of the sudden have this cold, after holding my cat who's been sneezing like crazy.  Heaven help me if I start coughing up fur balls!

Have you ever caught some sort of illness from your pets?