You can! When you do that, you essentially get a "Health Share" organization, wherein you pay cash for health care and they reimburse you. Google the term to see which ones are out there. They are typically much, much, much cheaper than traditional insurance.

The biggest drawbacks to such an approach are that you have to deal with the quirks and rules of your organization -- read the terms carefully -- and that you have to deal with the paperwork from the medical system yourself. That doesn't sound so awful until you experience it. You can find reasonable primary care as a cash patient -- direct primary care is everywhere now, and almost certainly what you want: a monthly payment for someone on call to deal with your issues. Urgent care is generally reasonable, too, being an up front single bill at a fixed price per type of issue. But Lord have mercy on you if you need to visit a hospital. The paperwork is stunning. A couple years ago, I went in for some observation and antibiotics for a couple of days, and came out with a dozen bills, all of which have to be called on, paid, negotiated, and some of which are possibly fake.