Is it possible to have purebred palominos?

Palomino isn't a breed of horse, it is a colour.

Answer 2:

As stated above Palomino is a color, not a breed. The Palomino horse registries are color registries, not breed associations. You get Palomino coloring by breeding a horse with the cream gene to a chestnut and you should get a palomino. Any breed that carries both the cream and chestnut colorings can produce a palomino.