No, I don't believe in reincarnation in anything. Once something has passed, it's passed on from this world.
BUT, I will say, that I believe in spirits reconnecting with the world. Perhaps a dog that you once loved dearly has passed. A new dog is brought in and perhaps the old dog gives the new one "tips" on how to console you.
Who is to say? Mine is just as unlikely as the next, but I take comfort in knowing that there are "feelings" that can't be explained. Walking through a closet and feeling a prickle on your neck when you walk by your mom's dresses. Seeing a dog that refuses to walk on linolium flooring walk on it and just stare at nothing in that same closet...