It's harder than you think, but once there's a good formula you can pull it off again and again. We do it at the office all the time, though with inanimate objects (computer boxes, stuff like that).

What's important is photographing your "placement sets" in nearly identical light - the dead giveaway is always that the lighting doesn't match. Then, when you've cut out your subject, use a 1-3 pixel fade on the selection before pasting to remove all evidence of a paste. I've never thought to try it with pets - though I was recently shown someone who does it regularly, and to be honest, it looked like crap.