They said that the website's woes show that the Obama administration and the federal government generally aren't capable of executing what the GOP says was an ill-advised program from the get-go.
"God only knows how much money they've spent, and it's a failure," Mitch McConnell, the Senate minority leader, said Sunday on CBS. "The government isn't going to be able to get this job done correctly."
On the other side, Democratic Sen. Jeanne Shaheen of New Hampshire sent a letter to Obama asking that the open enrollment period be extended past March 31, 2014. She also asked that he consider delaying assessment of a penalty to those who don't sign up for any health insurance before the so-called individual mandate kicks in.
Even Obama has been critical, insisting Monday that there's "no excuse for the problems." But he also said the problems should not amount to a blanket condemnation of the Affordable Care Act.
"Nobody's madder than me about the website not working as well as it should," Obama said, "which means it's going to get fixed."
Calls for resignation
Several top Republicans -- including 2012 vice presidential nominee Rep. Paul Ryan -- have called on Sebelius to step down due to the program's problems. The secretary skirted questions Tuesday about whether she'd step down, saying only that she works "at the pleasure of the President" and is committed to her job.
"I think my job is to get this fully implemented and to get the website working right," she told Gupta.
Billionaire investor Warren Buffett stood up for Sebelius in an interview with CNN's Piers Morgan on Tuesday night.
"I am a friend of Kathleen's, and I'm a friend of her when she's in trouble," he said. "I like Kathleen. I feel sorry for her in the position she's in. Obviously, it's a huge screw-up, but it will get worked out."
Sebelius refused to give a timetable Tuesday as to when the website will be fully operational, but she insisted it's improving every day.
"More people are having an easier time," she said, "and we intend to stay at this until we open the doors wide open."
And it's too early to call the rollout a failure, the health secretary said. There's still a long time for people to take advantage in person, by calling or by using the website during the open enrollment period.
When that six-month stretch is over, Sebelius said, people can better decide whether this part of Obamacare is a success or a failure.