Yes. Biting your nails is unhealthy for you. Biting your nails drastically increases your chances of getting sick (1). All the bacteria that you are touching gets collected under your finger nail and is released when you bite them. Just imagine how many things you have touched all day and how much of that will be transferred to your mouth. Also when you bite your finger nails you risk the chance of causing them to become infected. You might start seeing the skin near your finger nail turn red and it might hurting.
The other dangers of biting your nails is that they can sometimes become sharp. You might go to do something and accidentally cut your self. Or if you have other bad habits of picking your nose you might accidentally cut your nose.
Conclusion: Biting your nails is unhealthy and increases your chance of getting sick.
Sources 1- http://www.huffingtonpost.com/2014/08/18/nail-biting-bad-for-you_n_5675467.html