Bernie Sanders lost New York’s primary by a double-digit margin. Polls suggest his delegate gap with Hillary Clinton is likely to widen after five states vote on Tuesday. And Mrs. Clinton is increasingly turning her fire on Donald J. Trump, confident that she will soon lock down the Democratic nomination.

Still, Mr. Sanders’s supporters turn out, eager to hear his message but increasingly cognizant that their candidate’s political chances are fading quickly by the day.

“I hate to say it, but I feel like I’m putting my head in a bag and just crossing my fingers,” said Jessie Burnett, 39, a mother of three who lives in Tolland, Conn. “I’m throwing my full support behind him until he says he is out,” she said, calling his chance of winning the Democratic nomination “terrible.”

The increasing frequency of questions about whether and when Mr. Sanders might concede to Mrs. Clinton — or at least tone down his attacks on her — frustrates not only the senator but also many of his fans. As enthusiastic as ever, Mr. Sanders repeatedly tells packed crowds that they should encourage family and friends to cast votes for him and that his “political revolution” counts on their ability to get others involved.