When we're talking about rivalries between pets, then I think we're all well aware that what we see on the cartoons isn't exactly how it works. Cats chasing mice, Dogs chasing cats is what we're used to seeing but rivalries between different animals is only natural I think, and in the past I've had a rabbit that's chased a cat around the garden so there isn't any set rules as far as I'm concerned.
All animals are protective of their surroundings so introducing a new animal into the home is always going to be a challenge no matter what they are, but with a bit of time patience and effort, I do think that most animals can get over the initial rivalry, and their inquisitive nature will take over. As others have already said though, sometimes common sense as to prevail and if you're getting a couple of budgies for example, it's probably better not to let them fly around when you've got a few cats on the prowl!