I believe that reincarnation, as a repeated physical materialization, happens at all levels of evolvement of consciousness.
A soul is an entity that chooses to dedicate a fragment of itself to materialize in the physical reality, to experience, learn and grow. It can simultaneously send multiple fragments, not only successively (here "successively" isn't related to the "physical time").
As I understand this, the difference between man and animals is that man mostly mastered instincts, is dominated by emotions, develops an intellect, has an incipient intuition, while animals are dominated by instincts, develop emotions, have an incipient intellect. Men are distributed on a bell curve centered between emotions and intellect.
Anyway, as I see these, we don't reincarnate, but whatever incarnated as us keeps reincarnating, until it graduates. This is differently from saying "we reincarnate".