Minnesota is no-fault.
Auto insurers have the ability to contest no-fault claims two ways, by arguing either the treatment isn't reasonable or necessary, or by contesting whether it's related to the accident at all. Regular health insurance might fight with your doctor about the necessity of certain treatment, but in general its easier to get health insurance to pay.
As a practical matter I personally believe you are much better off getting as much no-fault insurance coverage and underinsured/uninsured coverage as you can reasonably afford. That is insurance that you buy to protect you, not some guy you kill with your car. An awful lot of people going around with just the minimum amount of liability coverage on their cars (might be $15-30,000 in some states) so when they drill you, you're on your own unless you have a lot of underinsured/uninsured insurance.
As a side benefit of purchasing a lot of underinsured/uninsured coverage, you will also have to buy a lot of liability coverage, which again isn't terrible, especially when you get to the stage in your life where you have assets to protect. A bad car accident with a couple of victims will blow through $500,000 in insurance coverage in no time.
