Mr. Moulton is in particularly bad shape: He has been in the race for more than two months but has not reached 1 percent in a single qualifying poll, according to the Times analysis. Aside from Mr. Sestak, who announced his campaign less than two weeks ago, Mr. Moulton is the only candidate in the 24-person field with this dubious distinction.

[Kamala Harris and Elizabeth Warren gained ground in a series of post-debate polls.]

The 14 candidates who have locked in a spot on the July debate stage by qualifying through both donors and polling, in alphabetical order, are: former Vice President Joseph R. Biden Jr.; Senator Cory Booker of New Jersey; Mayor Pete Buttigieg of South Bend, Ind.; the former housing secretary Julián Castro; Representative Tulsi Gabbard of Hawaii; Senator Kirsten Gillibrand of New York; Senator Kamala Harris of California; Gov. Jay Inslee of Washington; Senator Amy Klobuchar of Minnesota; former Representative Beto O’Rourke of Texas; Senator Bernie Sanders of Vermont; Senator Elizabeth Warren of Massachusetts; the self-help author Marianne Williamson; and the former tech executive Andrew Yang.

Who could get cut from the third debate?

An even larger group of candidates is expected to be left off the stage for the third debate in September. The D.N.C. raised the qualification thresholds for it significantly.

Candidates will need to have at least 130,000 donors and earn 2 percent support in at least four qualifying polls. And this time they will have to meet both standards to make the cut.

We surveyed the campaigns last month, and while most did not provide data on unique donors, a few did. Based on that information and what has been publicly released by the campaigns, we know that seven candidates have already met the donor requirement.

[Which presidential candidates raised the most money this spring?]

Five of those seven — Mr. Biden, Mr. Buttigieg, Ms. Harris, Mr. Sanders and Ms. Warren — have routinely eclipsed 2 percent support in the polls and will presumably meet the polling requirement with ease. They are all but certain to earn a spot on the stage in September. A sixth, Mr. O’Rourke, also has a track record of achieving 2 percent or higher in the polls, but a pair of post-debate surveys this week had him at 1 percent.

Mr. Yang’s campaign announced Monday that he had also exceeded 130,000 donors, making him the seventh candidate to do so. But Mr. Yang’s support in the polls has wobbled between 0 and 2 percent, and it is not clear whether he will achieve 2 percent four times in a two-month window.