Mrs. Clinton became sick. Several polls tightened to the margin of panic, with Mr. Trump overtaking her in surveys in Ohio and Florida. And even as Democrats hoped on Friday that Mr. Trump’s latest gambit — seeking to distance himself from his long history of “birtherism” — would backfire, there is a fear that no scandal can sink him.

A cartoon in The New Yorker captured it best: A woman sits in her psychiatrist’s office, perspiring in distress. The doctor scribbles on a pad. “I’m giving you something for Hillary’s pneumonia,” the caption reads.

Supporters of Mrs. Clinton have greeted the moment with varying degrees of alarm, according to interviews with dozens of them across the country.

They read warily about the health of her lungs and her swing-state field operations. They reassure one another by reminding themselves of President Obama’s two winning campaigns, which encountered similar fits of concern after Labor Day.

But even some zealous Clinton defenders have grown frustrated with their candidate, marveling at the prospect of her snatching defeat from the jaws of victory, for which some say they would never forgive her. The campaign’s decision last week not to acknowledge Mrs. Clinton’s pneumonia until two days after a diagnosis, once video surfaced of her stumbling out of a Sept. 11 memorial service on Sunday, has especially rankled.

“They kept it from us,” said Sonia Ascher, 74, a former campaign volunteer, sitting with her husband and son at a coffee shop in Portsmouth, N.H. “It was just another thing again, another mistake, which she really can’t afford right now.”