Most people who get dissatisfied with their insurance company do so because of a claim that was denied. Often, a person either does not understand their policy or is trying to commit fraud, so the company has to deny these claims. Any system, public or private, that is designed to filter out illegitimate claims from legitimate ones will occasionally deny claimants who should be paid and pay claimants who should be denied.
If I ever feel that my insurance company is not being fair to me, I can sue or switch companies. This is one reason why I oppose a single-payer health care system: I can switch insurance companies a whole lot easier than I can switch governments. If I am dependent on the government for health care and I don't like the way it's being handled, my best option is to wait until the next leap year and hope that a majority of the electoral college votes the same I do.