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The rocky rollout of the Affordable Care Act is the centerpiece of a Republican strategy to retake the Senate in November’s midterm elections. For months, operatives have battered vulnerable Democrats for standing by Mr. Obama when Congress enacted the law in 2010 and then promised Americans they could keep health care plans they liked.

That promise proved faulty when millions of consumers received cancellation notices last year saying their policies — which Mr. Obama repeatedly and emphatically said Americans could keep if they liked — did not meet Obamacare’s standards.

Mr. Obama decided to let people renew bare-bones plans for a few more years, but Senate Minority Whip John Cornyn, Texas Republican, told Fox News on Monday that the president’s credibility “has taken a body blow” since the episode.

Republican critics also say the administration has not provided a full accounting of its enrollment figures.

It is unclear how many customers were uninsured before they turned to Obamacare, or how many of them have paid their first month’s premiums.

Health and Human Services Secretary Kathleen Sebelius told an Oklahoma television station Monday that 80 percent to 90 percent of enrollees have paid, yet “lots of companies have different timetables for when their new customers have to send their first payment.

“You are not fully enrolled — you’re absolutely right — until you pay your premium,” she told News 9 KWTV.

It was the secretary’s clearest explanation to date. One Republican senator charged Sunday that administration officials were “cooking the books.”

“If we were cooking the books, don’t you think we would have cooked them in October and November?” Mr. Carney said Monday, referring to the woeful numbers last fall. “We could have saved ourselves a lot of pain.” covers about two-thirds of states, the ones that chose to let the federal government set up their exchanges.

Dave Boyer contributed to this report.