Dissatisfaction with work is indeed a known risk factor for back injury in the general adult population. A second, equally important factor is previous or current back injury. Repeated studies of college athletes show an injury rate of seven per 100 participants. Like the population at large, athletes are at greater risk for back injury if they've had a back injury in the last five years.
Dissatisfaction with the coaching staff or other team players does not seem to be connected to back injury. At both the college and professional levels, athletes are highly motivated. They have financial reasons to stay healthy and keep a positive mental outlook.