“All the Way,” the hit play about Lyndon B. Johnson’s first year in office that’s now in previews on Broadway, revels in the historical details of Washington politics, with large chunks of dialogue taken directly from the Congressional Record and Johnson’s voluminous phone transcripts.

But as close readers of Robert Caro or Taylor Branch will notice, the play sometimes veers wildly from the historical record it is trying to bring to life.

Several scenes, for example, feature Stokely Carmichael as the voice of radical black youth within the Rev. Dr. Martin Luther King Jr.’s inner circle — even though, in 1964, it was John Lewis who played that role (Mr. Carmichael, who later changed his name to Kwame Ture, didn’t depose Mr. Lewis as the head of the Student Nonviolent Coordinating Committee until 1966).

Likewise, the playwright, Robert Schenkkan, has Johnson weaken the voting rights provisions in the Civil Rights Act to make it more palatable to Southern Democrats. This never happened: the provisions were weak to begin with, but Johnson never touched them....