David Cutler, Amy Finkelstein and Kathleen McGarry write,

Standard theories of insurance, dating from Rothschild and Stiglitz (1976), stress the role of adverse selection in explaining the decision to purchase insurance. In these models, higher risk people buy full or near-full insurance, while lower risk people buy less complete coverage, if they buy at all. While this prediction appears to hold in some real world insurance markets, in many others, it is the lower risk individuals who have more insurance coverage. If the standard model is extended to allow individuals to vary in their risk tolerance as well as their risk type, this could explain why the relationship between insurance coverage and risk occurrence can be of any sign, even if the standard asymmetric information effects also exist. We present empirical evidence in five difference insurance markets in the United States that is consistent with this potential role for risk tolerance. Specifically, we show that individuals who engage in risky behavior or who do not engage in risk reducing behavior are systematically less likely to hold life insurance, acute private health insurance, annuities, long-term care insurance, and Medigap. [emphasis added]

Translation: some people lack insurance because they do not want it.

UPDATE: Mike Moffatt adds,

Most people lack the ability to accurately determine how much of a health risk they are or how bad their driving is. Being able to unbiasedly assess one’s own characteristics is one of the hardest things a person can do. So the information asymmetry likely runs the other way – your car insurance company likely is a better judge of your driving abilities than you are.