People with a Dog and a Cat in their house tell me that this is very common and happens all the time.
The Cat steals and occupies the Dog’s Bed.
So common, that Best Friend Services, an on-line memorial service for cats, made this ridiculously funny compilation video of Cats stealing beds from Dogs, and keeping them!
My favorite one is @0:48. Despite the dog dragging the bed around, the Cat bravely sits on the Bed, and gets a joyride out of it for free! ROFL.