“I don’t have a full answer for that,” says Mr. Paul, who says he believes ballot irregularities have chipped into his numbers in some places. He adds, “I think there’s some problem with always making sure this energy is translated into getting to the polls.”

Though the campaign says it still has some tricks up its sleeve, it now faces a hurdle just to place Mr. Paul’s name in contention for the nomination: a Republican convention rule requiring a candidate to have the greatest number of delegates from each of at least five states.

Mr. Paul said in an interview last week that he was “real satisfied” with the campaign. “You could always argue you wish you could do better, but we could have done worse, too.” He is already far ahead of his 2008 performance, including second-place finishes in seven states this year. But he remains in fourth place over all, with one-eleventh the delegates of the front-runner, Mitt Romney, according to an estimate by The Associated Press.

What Mr. Paul may have by the end of the nominating race is a valuable collection of delegates. So he could still play an important role at the August convention in Tampa, Fla. — especially if Mr. Romney fails to obtain 1,144 delegates to secure nomination on the first ballot, which is the hope, albeit perhaps fading, of Rick Santorum and Newt Gingrich.