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Sen. Barbara A. Mikulski, Maryland Democrat, said she wants the law to work but needs to see more clarity from the Obama administration.

“I think it’s very confusing about where you go,” she told Ms. Tavenner, referring to Marylanders seeking benefits. “We hear about the navigators and this and that, but I can tell you, people really don’t know.”

Centers for Medicare and Medicaid Services officials had to tamp down security concerns Tuesday, after a father in North Carolina said he logged onto the website this week and was able to view a notice containing a South Carolina man’s name, address and eligibility for subsidies.

“To be clear, as soon as it was reported to us, we put a fix in place to prevent it from happening in the future,” CMS spokeswoman Julie Bataille said on a conference call with reporters, noting it was the only reported incident of its kind.

Sen. Tim Scott, South Carolina Republican, said during Tuesday’s hearing he was troubled by the episode, and that the agency did little to help his constituent get his personal information removed from the site.

Supporters and critics of the law are making their cases through cherry-picked anecdotes, but GOP lawmakers say the number of problems cannot be ignored.

“It’s not what the president promised, and it’s not the kind of health reform Americans asked for,” Senate Minority Leader Mitch McConnell, Kentucky Republican, said Tuesday on the Senate floor. “So we should keep our focus where it belongs — on the real people getting hurt by this law.”