Mr. Mitchell bought a similar policy the same year from a broker who promised to find the best plan in Kansas at the best price. He was found to have invasive ductal carcinoma, a type of breast cancer, in early 2018. His doctor told him that he needed surgery, but the hospital where it was scheduled told him that he did not have insurance coverage.

According to the complaint, Health Insurance Innovations participated in a scheme involving Simple Health, another Florida company, whose agents sold them the flimsy coverage. Simple Health was recently shut down by the Federal Trade Commission after regulators accused it of being “a classic bait and switch scheme,” according to court filings.

The lawsuit claims Health Insurance Innovations spent millions of dollars funding Simple Health and was intimately involved in the scripts its brokers used to sell these policies. Customers were told that they were getting a P.P.O. plan, a traditional policy that allows individuals to go to the doctor or hospital of their choice. Many of the websites that consumers visited implied the policies being offered were from brand-name insurers.

The lawsuit quotes from the scripts being used by the brokers. “What’s the point of paying all that money every month if it’s not going to cover the most important things, right???” the broker asks. “This plan covers you from Day 1 …’”

Health Insurance Innovations says it was never involved in Simple Health’s activities. While the company said it had relied on Simple Health brokers to sell policies and collected the premium payments, it required all brokers “to provide clear disclosure of the information necessary for consumers to understand the policies they purchase,” according to its statement.

“Simple Health violated the trust of its consumers, its regulators and us,” the company said, emphasizing that it was not named as a defendant in the F.T.C. case and was cooperating with federal regulators.

The company said it cut its ties to Simple Health last year after the F.T.C. took action against the company.