“Nuts!” tells a quintessentially American story — one that elicits both wonder and horror.

Narrated in an irrepressibly chipper tone by Gene Tognacci, “Nuts!” opens with sepia-toned black-and-white images of goats copulating, roughly drawn and crudely animated. Along with the film’s title, and the soon-to-be-revealed fact that its subject, Kansas doctor John Romulus Brinkley (1885-1942), made his name and his fortune with an impotence cure achieved via goat testicle transplants, viewers are primed to settle in for an entertaining tale of made-in-the-USA bunco. This brisk movie is that, and more.

In this inventive documentary directed by Penny Lane (“Our Nixon”) from Thom Stylinski’s script, Brinkley is first seen as a barefoot teenager, begging for admission to Johns Hopkins University. This detail is gleaned from Clement Wood’s 1934 biography, which the movie’s credits name as a primary source.

“Brinkley would put goat testicles inside an impotent man, and nine months later, that man would call himself a father,” Tognacci says in a voice that suggests an affected straight face. The extent of the contrivance will set off multilevel credulity alarms in viewers paying close attention. Also curious is the movie’s seeming lack of interest in Brinkley’s inner life. These apparent defects are all part of the filmmaker’s genuinely cunning strategy.

Alternating with animated re-creations, archival footage and interviews with historians, Lane’s narrative, from Brinkley’s career roots in the once-one-horse town of Milford, Kan., grows to encompass a million-watt radio tower in Mexico, and more besides. The ever-taller-tale presented here documents Brinkley’s impact on early-20th-century pop culture. A Buster Keaton movie with a goat gland joke is cited, as is the fact that Brinkley’s radio station, besides being a showcase for his medical advice, was the first to feature all country music.

As the saga enters the 1930s, the animation and the archival footage shift to bright color, but the story grows more convoluted and bleak. The movie culminates in a cinematic coup de grâce bold enough to spin your head — one that gives the movie an entirely new dimension. No sooner does the twist sink in than “Nuts!” ends on a note of genuine tragedy.

(At the Tivoli.)