It is possible to cut a nerve from self-harming; however, it depends on the depth of the wound.
I've cut that deep, mainly on my arms and thighs. The first time I cut that deep was completely accidental. There was an instant stabbing pain when I did, not terrible pain but pain. I didn't tell anyone, just did my best to cover the wound using worn shirts and tank tops.
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To cut a nerve, you would need to cut the hypodermis, which mainly consists of fat. You can tell you've cut that deep when the wound gapes opens and you see yellow. They also usually bleed more and for longer.