Does anesthesia sometimes wear off before the surgery is complete?

No: General anesthesia is maintained throughout the case, and is only turned off at the end of the case. Regional anesthesia is usually limited to a certain timeframe, but is chosen based upon how long the surgery should take. If it wears off then the anesthesiologist can supplement with IV medications.

Should not happen: This should not happen if your anesthesiologist is alert except in very rare cases. The most common situations are emergency surgeries or very sick patients undergoing surgery who cannot tolerate very much anesthesia.