Is it possible to die by puncturing the veins in your wrists?

Probably not: A puncture of the veins on the wrist is probably not fatal. It will probably seal and stop bleeding.

No.: The wrist has both arteries and veins. Veins are low pressure and arteries are high pressure. If veins are cut, the blood will usually trickle and stop with direct pressure. If the artery is cut, the blood will squirt and sometimes even direct pressure will have trouble controlling the bleeding. Venous bleeding at the wrist should be able to be stopped by pressure and not lead to death.