I think they do. Just like dogs. I've caught Dobby dozing off and watch his little mouth going a mile a minute like he was enjoying something good. Then he'd snap out of it.
Ha, too funny Karen. Same here, I've caught Toby sound asleep a couple of times.
He'll be flopped on his side, all of a sudden his legs will start going as if he's running around, then he'll stop and his mouth will start going like he's chewing. Poor boy must be dreaming about dandelion greens and clover! Darn winter is too long for him