Some cats have the bad habit of biting their nails same as in the human beings. It is mainly because of anxiety, stress tension and conflict. Same can be the reason for the catâ€™s bad habit of biting nails. Cats do not chew their nails all the time they do it more or less continuously.
You must have noticed that some people start biting nails when they are suddenly stopped of doing a particular task like on red light signal some people bite their nails because they want to move but compulsively they have to stop which shows the conflict and they displace into grooming this is called compulsive disorders. Same happens with the cat they start grooming themselves whenever they are in an anxious situation. So some cats often start grooming their coat but sometimes they go for their nails. It totally depends on which neural pathway was most active when a cat was kitten.
Some says that biting nails is common in cats and there is nothing much to worry about because when nails grow long and starts shedding cat will bite their nails to remove the old layer of the nails as the nails in cats not only grows in length but also in the layers. But if the cat is biting her nails to the quick then you must take your cat to veterinarian for consultation as it may not be normal.