WASHINGTON — Kathleen Sebelius, the secretary of health and human services, gamely defended the problem-plagued rollout of President Obama’s health care law on Wednesday and tried to explain the cancellation of hundreds of thousands of individual insurance policies.

In three and a half grueling hours of testimony before a House committee, Ms. Sebelius apologized for the missteps and problems in efforts to carry out the president’s most important domestic initiative.

But nothing Ms. Sebelius said at the hearing could overcome the stark message displayed on a large video screen that showed a page from HealthCare.gov: “The system is down at the moment. We are experiencing technical difficulties and hope to have them resolved soon. Please try again later.”

That message undercut Ms. Sebelius’s statements that the federal online insurance marketplace was improving every day and would be fully functional and able to accommodate millions of users by Nov. 30 — about a month before the deadline for buying coverage on Jan. 1.