When the question arose, Mr. Obama’s advisers decided that the assertion was fair, interviews with more than a dozen people involved in crafting and explaining the president’s health-care plan show.

But at times, there was second-guessing. At one point, aides discussed whether Mr. Obama might use more in-depth discussions, such as media interviews, to explain the nuances of the succinct line in his stump speeches, a former aide said. Officials worried, though, that delving into details such as the small number of people who might lose insurance could be confusing and would clutter the president’s message.

“You try to talk about health care in broad, intelligible points that cut through, and you inevitably lose some accuracy when you do that,” the former official said.

The former official added that in the midst of a hard-fought political debate “if you like your plan, you can probably keep it” isn’t a salable point.