Kinda.
The Australian system is a blend of the US system and the British system. We allow private insurance and private working doctors and at the same time have a public health system much like the British. We do not have practices that are owned and run by the private insurers, that the US does have (HBOs I think they call them?)

We have long queues in the public system and the private system isn't going too well either. Since the private system can't really discriminate on customers (charging more for the unhealthy customers) and having a public system where the healthy go (why have insurance if you aren't gonna get sick and there is a system that will look after the worst injuries?) the private system has a tradgedy of the commons problem.

Hope that helps