It depends on where the neck was broken.
If you break the C3, C4, or C5 vertebrae, the nerves that come out of the spinal cord in that area enervate the diaphragm and control breathing. Breaking C3, 4, or 5 will discontinue the patient's ability to breathe on his/her own.
Most breaks result in paralysis, usually from the injury site downwards, resulting in quadriplegia.
However, in rare cases the spinal column remains aligned until further trauma or movement. This is why after an accident with severe trauma in which the neck may have been injured, it is important to protect the neck and spine until doctors can be certain that the spinal column is not compromised. It's happened before that someone in a car accident thinks they're fine and gets out of the car, only to twist or bend later on, misplacing the break in their spinal column, and resulting in paralysis.
If you think you or someone you know may have broken or injured their neck, even if it's been a little while, go get checked out immediately.