I live in Ontario, Canada where we have a single payer health care system. But this system, as good as it is, does not cover everything. So many working Canadians also get supplemental health care insurance through their employer (it is a sought after benefit, one that unfortunately I do not have). This supplemental insurance covers things like dental, physiotherapy, massage therapy, or a private hospital room.
My thesis is that for these supplemental services insurance actually serves to inflate the costs of these services and allows providers to earn excessive salaries.