This often happens when the skin under your nail becomes thin and dry. It is quite a common problem that many of us encounter. As you said, though the nail grows back, it might get cut in the same place again and again. This happens because it is where your skin meets the nail. Until that spot, your skin must have got dried making a gap between the skin and nail.
To solve this problem, you can moisturize that part of the skin. Apply a moisturizing lotion or Vaseline for a few nights and wear latex socks on, to help it get healed. The dryness of the skin should be treated first, whereas the skin gets healed on its own slowly.