Yes a nerve can be hit when a nurse is drawing your blood especially if they are not quite sure what they are doing.
I've had nurses before that could barely find the vein in my arm to take the blood from and had to stick me with a needle at least 8 times before they got it.
Luckily they didn't hit a nerve in my arm while drawing blood but I've heard it's happened to other people.
If I go to a doctor and they cannot draw the blood on the first or second try I leave that doctor and find another nurse.
Some people have experienced nerve damage as a result of the nurse hitting the nerve but sometimes they aren't harmed other than the nerve getting hit.
Most doctors are pretty competent and won't hit a nerve but there are some out there that could hit a nerve.