Can a hand fracture make your hand go numb?

Possibly: It depends on which bones fractured, and how they broke. Sometimes nerves can be injured also. If you have injured yourself, please see a doctor asap.

Yes if excess : Swelling either put pressure on nerve and blood supply or makes a cast or splint too tight or bone or joint put pressure on a nerve . Numbness after a fracture means seek medical attention right away.