When 911 should be called, would depend upon how life threatening the situation is. If the situation isn't life threatening then administering first aid may be enough to safely transport the injured person to a proper medical facility. Where I would draw the line is if the person is having breathing problems, bleeding uncontrollably, is showing signs of heart attack or stroke, or any other life threatening issue. If it is life threatening, 911 should be called immediately.
If the situation isn't immediately life threatening, taking a moment to consider a plan of action could be prudent. For some, the cost of emergency care may be a factor in the decision to call for help. For instance, if you broke a leg while back country camping, the decision to call in a helicopter may be one you would take a few moments to consider. If you had a hiking buddy to help you out, plenty of food and water, not close to going into shock, and the skills to adequately splint the injury, then the decision to call for help would have to be a very personal one. There are personal and financial consequences with the above choices.
No amount of money is worth your life, so if it is life threatening, then please call 911.