My cat'll hang around you and never let you leave her sight if you're not feeling well. She'll follow you and make sure your okay. It's very comforting, actually, unless you've got a serious case of the runs and enjoy privacy while you *ahem* "take care of business". I know that when you're feeling upset and crying she'll always be there, purring as loud as she can (which isn't very loud) and rubbing you so that you can love her.
From what I've heard and read this is actually a very common phenomenon among many pets, like dogs and cats. They somehow seem to know if you're sick, or in physical or mental distress. For instance, there have been many cases of dogs knowing someone is about to have a seizure before they actually have one, and taking them to a safe place to have it. I'd like to know how and why they do this.