The following passage was taken from Sri Chinmoy’s book entitled Death and Reincarnation:
Question: Can we reincarnate as an animal?

Sri Chinmoy: The world is evolving, progressing; we are going towards our goal. In the march of evolution, once we have accepted the human life, we do not go back to the animal kingdom. Now every rule has exceptions, but these exceptions are very, very rare. If an individual wants to re-enter into the animal life, we have to know that that particular individual is not at all progressive.

     If a person has just entered into the human life, if it is his first or second incarnation as a human being, and if his propensity towards the animal life is still very strong, that is to say, if his impure desires were not fulfilled in his past incarnations in the animal life itself, then that human being may go back to the animal life. What happens is that the time comes when a part of his animal life becomes fully awakened and wants to enter into the human life. But there is some part of him which is not fully prepared for the human life. Still, the soul comes to the human life in spite of being imperfect, in spite of not being fully prepared for it, and then is later compelled to return to the animal life because the unillumined part in him has not been transformed and has most powerful animal tendencies and desires.

     Another reason is that when the lower vital – the unsatisfied, unfulfilled vital – of an ordinary human being is causing him unbearable suffering and pain, if he feels that in the animal kingdom there is more opportunity to enjoy the lowest vital, the gross physical enjoyment, then that particular soul enters into the animal kingdom for a few months so that it can have the grossest enjoyment very freely. It is not for a year or two, but only for a very short period. When the enjoyment is over, when the necessary experience is over, the soul enters into the human world again. Man has transcended the animal, but not totally. Inside us there is an animal. With our conscious prayer and meditation we are trying to transcend the animal fully. By the time aspiration begins, the soul is far too highly developed ever to return to the animal kingdom.

– Sri Chinmoy, Death and Reincarnation

    The soul does not go back to animal condition; but a part of the vital personality may disjoin itself and join an animal birth to work out its animal propensities there.
    But the soul, the psychic being, once having reached the human consciousness cannot go back to the inferior animal consciousness any more than it can go back into a tree or an ephemeral insect. What is true is that some part of the vital energy or the formed instrumental consciousness or nature can and very frequently does so, if it is strongly attached to anything in the earth life.

Sri Aurobindo, From “A Practical Guide to Integral Yoga“.