But strangely, this film, which claims to be the first ever to present McVeigh in his own words, blunts its impact by relying on stagy computer graphics, and even an actor whose looks are digitally altered, to re-enact McVeigh’s movements. Scenes of this domestic terrorist in shackles during a prison interview or lighting a fuse inside a rented Ryder truck look neither real nor completely fake, but certainly cheesy: a violent video game with McVeigh as a methodical, murderous avatar.

Documentaries increasingly use technology, often to good effect. The History channel, which used to rely heavily on quaint, period-costume re-enactments (a shot of a quill pen writing on parchment, a tableau of soldiers firing muskets at close range), is expanding its virtual reach. “America: The Story of Us,” a six-part series about the United States that begins on Sunday, is visually thrilling but in a sensible, instructive way: computer wizardry, for example, peels a map of Manhattan today back to what the terrain looked like more than 200 years ago when the towers and strip mall of Kips Bay were meadows stormed by the British in 1776.

“The McVeigh Tapes” sometimes casts aside real material in favor of a faked re-creation. One catalyst for McVeigh’s terrorist act was the April 19, 1993, assault by federal agents on the Branch Davidian compound in Waco, Tex., that left 75 people dead. McVeigh visited Waco in March 1993, when it was under siege by government troops. By chance a student reporter, Michelle Rauch, interviewed him as he lounged on the hood of his car.

“He was very unassuming,” Ms. Rauch, who is still a journalist, recalls in the film. She explains that she didn’t know what to make of his words: “People need to watch what’s happening and heed any warning signs.” Ms. Rauch says she didn’t realize until a year after the bombing that her interviewee was Timothy McVeigh. “Well, when I went back and read that in my article, it gave me chills,” she says.

Most documentaries would show her notebook or a clipping of the actual newspaper article  another odd scrap of fate in a real-life puzzle. This one, more focused on ersatz re-enactments, doesn’t.