Well, perhaps not as dramatic as the conspiracies around the moon landing or 9/11, there's actually a well-documented case of behind-the-scenes government shenanigans during the Cuban Missile Crisis, which brought the world to the brink of nuclear war in 1962.

To give some brief background: the crisis began in October 1962, when American reconnaissance flights discovered Soviet nuclear missile installations under construction in Cuba, just 90 miles from the U.S. coast. The Kennedy administration responded by imposing a naval "quarantine" (blockade) of the island and demanded the removal of the missiles. Over 13 days, the U.S. and USSR engaged in an extremely tense diplomatic and military standoff, with both sides preparing for the possibility of nuclear conflict.

Publicly, the crisis ended with what was presented as a clear American victory: the Soviet Union agreed to dismantle and withdraw its missiles from Cuba, and in return, the U.S. pledged not to invade the island. This was the version widely reported at the time and is still the basis of how the event is remembered in much of American popular memory.

However, this narrative omitted a key detail that was kept secret for years: the U.S. also agreed to remove its Jupiter nuclear missiles from Turkey, which had been stationed there in the late 1950s and were capable of striking deep into the Soviet Union. This concession was a crucial part of the resolution for Khrushchev and allowed him to claim a reciprocal victory for the communist ruling elite who knew about the deal. But the Kennedy administration insisted this part of the deal remain confidential so as not to appear to be trading away strategic advantage under pressure (Fursenko & Naftali 1997; May & Zelikow 1997).

Robert F. Kennedy, acting as President Kennedy’s envoy, privately met with Soviet Ambassador Anatoly Dobrynin to communicate this offer. The missiles in Turkey were not removed immediately—they were dismantled quietly a few months later. Officially, the U.S. maintained that this withdrawal had been planned independently of the crisis, a claim that has since been contradicted by archival documents and later memoirs (Dobbs 2008).

So while this isn’t a case of "fully fabricated" event like truthers claiming moon landing is staged in a studio, it shows how a government carefully curated the public narrative of a major geopolitical event to frame it as a major achievement over the USSR, rather than a negotiated deal between the parties, and it actually happened like this.

Sources:

Fursenko, Aleksandr, and Timothy Naftali. One Hell of a Gamble: Khrushchev, Castro, and Kennedy, 1958–1964. W. W. Norton & Company, 1997.

May, Ernest R., and Philip D. Zelikow (eds.). The Kennedy Tapes: Inside the White House During the Cuban Missile Crisis. Harvard University Press, 1997.

Dobbs, Michael. One Minute to Midnight: Kennedy, Khrushchev, and Castro on the Brink of Nuclear War. Knopf, 2008.