The phrase "to tear the nail" usually means to remove or pull out a nail forcefully or accidentally, damaging or breaking it in the process. Full definition
Is there a surplus of similar appointments, like an increase in torn nails, perhaps? (lifelearn.com)
One of my own dogs once tore a nail and had to have it completely removed. (dogs.lovetoknow.com)
These baby nail clippers come in different shapes and sizes and are designed to cut your baby's nails safely without tearing the nail or cutting too deep and causing a minor bleed. (babygaga.com)