The political dangers were clear to everyone in the room: Vice President Joseph R. Biden Jr.; Kathleen Sebelius, the health secretary; Marilyn Tavenner, the Medicare chief; Denis McDonough, the chief of staff; Todd Park, the chief technology officer; and others. For 90 excruciating minutes, a furious and frustrated president peppered his team with questions, drilling into the arcane minutiae of web design as he struggled to understand the scope of a crisis that suddenly threatened his presidency.

“We created this problem we didn’t need to create,” Mr. Obama said, according to one adviser who, like several interviewed, insisted on anonymity to share details of the private session. “And it’s of our own doing, and it’s our most important initiative.”…

Inside the West Wing, where junior researchers monitor Twitter and other social media, officials knew the political controversy had moved beyond the broken website. Now it was about a broken promise. But for Mr. Obama, the mounting criticism was more than political. It felt personal.

“He was uncomfortable,” one senior adviser said. He hated the idea that so many Americans had received cancellation letters from their insurance companies and were angry because “of what the president had said — that this wouldn’t have happened.”