Credibility gap is a term that came into wide use with journalism, political and public discourse in the United States during the 1960s and 1970s.
1975: "The Credibility Gap" - Tom Snyder - UFOs - and CIA ... by VortexTech
At the time, it was most frequently used to describe public skepticism about the Lyndon B. Johnson administration's statements and policies on the Vietnam War.
Lyndon Baines Johnson, often referred to as LBJ, was an American politician who served as the 36th President of the United States from 1963 to 1969, assuming the office after serving as the 37th Vice President of the United States under President John F. Kennedy, from 1961 to 1963.
The Vietnam War, also known as the Second Indochina War, and known in Vietnam as Resistance War Against America or simply the American War, was a war that occurred in Vietnam, Laos, and Cambodia from 1 November 1955 to the fall of Saigon on 30 April 1975.
Doris Kearns Goodwin Explains Trump's Credibility Gap by The Late Show with Stephen Colbert
Today, it is used more generally to describe almost any "gap" between the alleged reality of a situation and what politicians and government agencies say about it.