It is possible. A recent study from Japan showed cellular changes one week after injury. In this case, dogs were used as the study model. The dogs had disc herniation with or without pressure on the spinal nerves. There was fiber damage and swelling present inside nerve cells.
Those dogs with both disc herniation and spinal nerve pressure had much greater cell changes. The authors of the study think that a combination of mechanical pressure and chemical irritation causes more nerve root injury. Pressure from the disc by itself, or irritation of the nerve tissue alone had smaller changes.
Studies with humans will be the next step in understanding how these injuries affect the body.