I haven't had this one come up yet but judging from the super rich people I know a little about I don't think it happens much.
My idea is the U.S. government even under Obamacare did not ration healthcare on a daily basis. If I REALLY wanted an MRI because I was playing football or baseball and tweaked my widget, Washington DC generally did not make my insurance company pay for it but DC also did not prevent my employer or me from paying for this MRI on my widget.
It seems my ability to pay for that MRI, or a second filling in two years plastic surgery or whatever is the limiting factor so capitalism rations my healthcare. In essence if you are mad because your insurance doesn't cover something you want BIGGER government to force the private company which is there to make money to pay for something.
(We'll exclude mass casualty events from this conversation. Triage techniques are a whole other topic)
When does rationing start? Simple. When resources become scarce enough to demand it. When that point comes, that's a little harder. For that, I think we can turn to Europe for clues. Or, even closer to home, Canada.
As you point out, economics have already started rationing based on who can pay. When costs get to the point where government can no longer borrow enough to pay, the rationing will commence. Of course it will be called something else. Just look at Social Security. Every so often, congress debates, and inevitably, raises the retirement age. Something similar will happen with healthcare. It will be billed as a "measure to ensure the greatest benefit to the largest number of people", an unfortunate effect of that will be that some people, those seen as not viable contributors to society, or above an arbitrary line of wealth/income, will go without, unless they can pay. In short it will be the old and the dying that get left out first. Grandma, and the terminal kid down the street.