I'm only familiar with our system in Aus. We have a two-tier system. The single-payer scheme run by the federal government authority provides universal health care. But there is a second-tier of private practice which you can access if you have insurance or you can pay for it. It doesn't give "extra" medical care as such but it does mean that you can get elective surgery (as an example) quicker (this doesn't affect the situation where surgery is simply required right now, if that's the case you get it).
The two-tier system in Aus upsets the more ideological amongst us but in reality it works okay because it does relieve the pressure on the so-called "public" system. I don't have much time for ideology when it comes to health care, I'm more interested in effective health care delivery.