Minnesota's health exchange is an example of a state-run exchange that is performing well, the newspaper said.
Robyn Skrebes of Minneapolis, who is 32 and had lacked insurance, said she signed up for health coverage in about two hours on Monday using the state-run website, MNsure. She said she purchased a policy for $179 a month, before tax credit subsidies, and also got Medicaid coverage for her 2-year-old daughter, the Times reported.
"I am thrilled," Skrebes said. "It's affordable, good coverage. And the website of the Minnesota exchange was pretty simple to use, pretty straightforward. The language was really clear."
Other states with state-run exchanges that seem to be operating well include California, Connecticut, Kentucky, New York, Rhode Island and Washington, the Times said.
Meanwhile, problems continue to dog the federal health exchange website, which serves people in 36 states, although they are occurring less frequently, according to published reports.
And the Washington Post reported Wednesday that big insurance companies, state health officials and Democrats friendly to the Obama administration said they'd been warning federal health officials for some time that the federal exchange had serious shortcomings. The White House proceeded with the Oct. 1 launch despite those warnings, the newspaper said.
Rep. Robert Andrews, D-N.J., played a major role in helping to get the health-reform law known as the Affordable Care Act passed in 2010. The online exchanges -- the federal site and the state-run ones -- are key components of the law, sometimes called Obamacare, which is designed to bring health insurance coverage to millions of uninsured Americans.
Andrews told the Post that he told White House officials early this summer that he'd been hearing from insurers that the federal online system was flawed.
"Nothing I told them ever surprised them," Andrews told the newspaper. "The White House has acknowledged all along something this massive was going to have implementation problems."
One senior administration official, who requested anonymity from the Post, said the Obama administration expected some problems with the launch of the federal health exchange, "but we had a lot more traffic than we thought, and so discovered problems managing that load."