They're telling the truth but it sounds absolutely insane.

1) Howard Hamlin from Better Call Saul. Jimmy and Kim enact a scheme in order to get the Sandpiper case settled. To do this, they convince everyone that Howard is addicted to Cocaine. While they did absolutely sabotage him, everyone around him chalks it up to his nonexistent addiction.

2) Fives from Star Wars: The Clone Wars. Fives learns of the inhibitor chips in all the clones and finds a massive conspiracy which would end with Order 66. Chancellor Palpatine admits to the conspiracy in private knowing that no one would believe him. This, combined with him saying that the proof was in his head, results in everyone thinking he simply snapped.

3) Hank Hill from King of the Hill. In the episode "Hank's Dirty Laundry" Hank goes to great lengths to prove he did not rent any pornographic films from Arlen Video even going so far as to produce from said films in small claims court. It is implied at the end of the episode that Bill may have used his name to rent the films, thereby incurring late fees that prevent Hank from getting a new dryer. While renting pornos is extremely out of character for Hank, the lengths he goes to makes his friends, family, coworkers, the small claims judge, and even the audience wonder why he didn't just pay the $40.