The case made — convincingly — by the documentary “The Newburgh Sting” is that the four upstate New York men convicted of plotting to bomb synagogues in the Bronx and destroy airplanes in 2009 were the fall guys in an elaborate, cinematic performance orchestrated by the F.B.I. The film, showing Monday on HBO, offers a competing narrative to the one presented at trial and in the news media by the government. So far, the government’s story is winning: The men are all serving 25-year sentences in federal prisons.

Much of what we see in “The Newburgh Sting,” directed by Kate Davis and David Heilbroner, was provided by the F.B.I. The directors make extensive use of government surveillance video shot during the year in which an informant, Shahed Hussain, presented the plot to the four Newburgh men, offered them $250,000 and supplied them with the weapons to carry it out. It’s also highly selective use, of course, leaving out the most inflammatory anti-American and anti-Jewish rhetoric of the lead suspect, James Cromitie. (He later disavowed those statements, saying they were part of his own con to convince Mr. Hussain of his terrorist bona fides.)

In a broader sense, though, the film builds a credible circumstantial case for the entrapment defense. It does so through the standard techniques of true-crime documentary: interviews with sympathetic, cogent relatives and acquaintances of the four men; an evocative depiction of life in a depressed Hudson River Valley town; the canny juxtaposition of just-folks black and Muslim Newburghers with a succession of suited, white law-enforcement officials, politicians and television talkers, often saying things we can see aren’t true.

Two characters stand out. Alicia McWilliams, aunt of David Williams, one of the conspirators, is magnificent in her anger, despair and dark humor. “How could you come in our community and prey on these damn fools?” she snaps. “David should have only got five years for not having common sense.”