Senate Majority Leader Harry Reid, D-Nev., kept his distance from a measure by Sen. Mary Landrieu, D-La., who faces re-election next year, that would force insurers to reinstate canceled policies.
"There are hundreds of bills introduced every week, and we have to sort through those that have opportunity to be voted on," Reid said.
On the defensive about the law, Democrats have started trying to refocus Americans on its benefits.
The law requires most Americans to have health coverage by the start of 2015 or face fines. Middle-class people who don't get health insurance at work will qualify for federal subsidies for the private coverage they buy. More lower-income people will qualify for Medicaid in states that have agreed to expand that federal-state health care program for the poor.
"The real train wreck is what people are experiencing every day because they can't afford care," Sen. Tom Harkin, D-Iowa, said Tuesday at a hearing of the Senate Committee on Health, Education, Labor and Pensions, which he chairs.
At that hearing, Marilyn Tavenner, who runs the agency most directly involved in implementing the health care law, said the website was being improved.
"We obviously underestimated demand," said Tavenner, who heads the Centers for Medicare and Medicaid Services. She added, "We obviously had more bugs than we realized."
AP Special Correspondent David Espo and AP reporters Ricardo Alonso-Zaldivar and Josh Lederman contributed to this report.
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