I have private health insurance and intend to keep it. If I was in my twenties again I would self insure by saving and investing the premium each year. It really depends on where you live, what resources are available to you and what you want to insure for. There are a wide range of serious illnesses that can be treated in the public health system perfectly well, that are generally NOT treated or not treated WELL in private hospitals, and if you are insured and in a public hospital it makes not one jot of difference by and large. That being said being a in a public hospital is like being on a bus...it will reliably get you there in the end, but you won't go the most direct route and there will be stops and starts along the way.
That being said elective and even urgent surgery sometimes, can be treated well and quickly in the private system in comparison to the public. Although I'm pretty healthy, and without going into details ( secret women's business) I've had 6 surgeries, 3 of them urgent in the private system. And probably as many day only minor procedures. Access in the public system would have taken forever.
If you have a large comprehensive private hospital nearby, then you will get better value for money than if you don't. My parents used one in Sydney that had an emergency department and this was invaluable over the years and a vastly more pleasant experience than public hospital EDs. Again if you have a large comprehensive private cancer centre you can access this will be useful, but public cancer centres are also pretty good.
I have bad teeth and bad eyes, so generally break even on my extras cover without anything else. Throw in a couple of ambulance trips and some physiotherapy in recent years and I'm not losing.
However its not a clearcut question and many folks are happy not being insured and I wouldn't argue with this either.