This is very likely to be completely normal feline behaviour. It is part of the self-grooming process. Your cat is attempting to remove the outer layer of one of her claws which they also do by scratching a scratching post or a tree or a fence outside. You might see the outer layer of a claw lying around the house from time to time. Sometimes people think this is an actual claw but they are not. They do look a bit like a claw. The purpose? To sharpen the claw by exposing fresh claw underneath.
This process has nothing to with a cat trimming her own nails. It is simply part of body maintenance, a bit like humans cutting their nails, shaving, combing their hair and those sorts of activities. They use their tiny incisor teeth at the front of their mouths to do it.
It appears to me that they bite at the base of the nail with their incisor teeth (both upper and lower jaw) and pull away from the toe in an attempt to pull off the loose outer sheath of their claw. It doesn’t always work because the sheath is not loose but cats seem to do this automatically. It is possible that cats sense when the claw needs to be maintained and the outer sheath removed. This may be because they have an internal cat maintenance clock as is the case for grooming their fur or because they can feel the outer sheath becoming loose.
I don’t know of any other instance in terms of health which results in a cat biting her nails unless, rarely, there may be something on her paws, specifically her paw pads, which invites her to nibble at them. But the specific behaviour referred to of biting nails is very obvious as I’ve described. If the actions are not as described and you suspect an injury to the paw or contact dermatitis or something like that please take her to a vet for a consultation.
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