The classic case was the Iraq War, where the official line was that there was a massive conspiracy involving Saddam and Weapons of Mass Destruction. Opponents of the war argued that that theory was fake, and that Western Governments were involved in a conspiracy to bring about a war.

And in the end it turned out that the conspiracy theory about Saddam, which sounded very plausible to many, was a fake, and the conspiracy theory about Western Governments, which on the face of it sounded crazy to many people, was the truth.