Is it possible to survive after having a broken neck?

Yes. If you get it immobilized in time, and the vertebrae don't injure your spine, you may do well. Even if you are paralyzed, you can survive, although you may need a breathing machine (ventilator) to live (depending on how high/severe the injury is).
Yes. The survival following a fracture of the cervical depend on the level of the fracture and importantly whether there bony fragment displacement impinging the spinal cord. Rapid immobization, body temperature cooling, high dose i. V. Steroids all play a part in survival and potential for neurological recovery. Even paralysis such as tetraplegia or paraplegia can live a long and productive life.