My brother didn't get something he was supposed to get from the army (he's a vet), my mother raised hell with our senator, the senator raised hell, and my brother got what he deserved. I have no reason to think that a National Health Care service would be any different.
It's one of the ways it works in Britain, at least - go to your MP and make a fuss. The other is a built-in system of appeals and inspections, which however has its flaws. Then there is the press, which is the nuclear option; people who get seriously taken to task in the press, in those stories that get to America, most often have to resign. In Italy there is a fourth and much better way, that is "Administrative Justice" - a system of independent and very powerful courts that adjudicates in any complaint against public bodies.