Can acupuncture cause nerve damage?

Not really. Theoretically and in reality it does not happen because nerves are very slippery and movable, and the acupuncture needles flexible. Because of that it is virtually impossible to pears a nerve. Of cause care is being taken that nerves are avoided. Acupuncture points are located along the fascias (very thin transparent tissues enveloping the muscles) and usually away from the nerves.
Very rarely. Acupuncture can rarely cause damage to superficial, usually sensory nerves, should the needle injure the nerve. There have been several reports in the neurologic literature regarding these events. All the patients recovered without long term sequelae. Given the number of individuals seeking acupuncture, the chance of nerve injury is very slim.
Uncommon. Nerve damage is one of the possible problems that can occur from acupuncture - but it is rare. A gentle needler would be well aware of the pain caused if too close to a nerve. However, major acupuncture points are in the vicinity of nerves. We use the electrical field surrounding the nerve to obtain chi.

Related Questions

What exactly is the "nerve damage" that acupuncture can cause?

Nerve damage. Acupuncture does not cause nerve damage. Rumors of nerve damage are a myth.
Should be none. Acupuncture needles are not stuck into nerves so there should be no nerve damage.
Uncommon. Needles are to be placed in close proximity to acupuncture points so as to contact the electric field (or qi). Needles are not to be placed in nerves. It would be possible for a sloppy needler to rapidly insert a needle into a nerve which would cause inflammation and thus injure the nerve. This does not happen often (if it did people would be very reluctant to see the acupuncturist back).