It definitely depends on the combination of pets you're going for. Our dog was so jealous of a new baby that he went to live with my grandparents - obviously we couldn't get rid of my baby sister! But I've had cats behave in all sorts of ways - our old tom cat (neutered) took to the pregnant stray we took in and would curl up with her, but was terrified of her kittens. Meanwhile an elderly female cat we had never really learned to like the stray, they tolerate each other but won't cuddle or get too close together.
I did know someone who took her pet rabbit to the shelter to "choose" a new friend when her original companion died and it seemed to work really well! They came home with the fanciest looking long haired albino rabbit, I do wonder if her bunny chose based on looks