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The film follows Theodore Honey (James Stewart), a highly eccentric "boffin" with the Royal Aircraft Establishment. A widower with a precocious young daughter, Elspeth (Janette Scott), Honey is sent from Farnborough to investigate the crash of a "Reindeer" airliner in Labrador, which he theorizes occurred because of a structural failure in the tail caused by sudden metal fatigue. To test his theory in his laboratory, an airframe is continuously shaken in eight-hour daily cycles.
It isn't until Honey is aboard a Reindeer that he realizes he himself is flying on one such aircraft and that it may be close to the number of hours his theory projects for the fatal failure. Despite the fact that his theory is not yet proven, Honey decides to warn the passengers and crew, including actress Monica Teasdale (Marlene Dietrich). After the Reindeer lands at Gander Airport an inspection clears it to continue on. He takes drastic action to stop the flight by raising the undercarriage while the aircraft is still on the ground, lowering the aircraft to its belly and damaging it. Shocked by the act, some people demand that he be declared insane to discredit his theory.
Teasdale and flight attendant Marjorie Corder (Glynis Johns) both take a liking for Honey and Elspeth, who is lonely and isolated from her schoolmates. Teasdale speaks on his behalf to his superiors, while Corder, seeing that he is decent but disorganized, decides to marry him.
During a hearing in which his sanity is questioned, Honey resigns but continues trying to prove that his mathematics are sound. In the laboratory, the time he predicted for failure passes without failure. The Reindeer he disabled is repaired, but after landing from a test flight the tail falls off. Shortly afterward, the same thing happens to the test frame in the lab, and Honey discovers that he failed to include temperature as a factor in his calculations.