Ray Liotta, along with the usual preparation of watching Frank Sinatra's performances and films, also watched Nixon (1995) starring Anthony Hopkins. Liotta was inspired by Hopkins because Hopkins did not resemble Richard Nixon but convincingly portrayed him.
On election night, Frank Sinatra is talking on a phone. The receiver is clearly connected to the phone wire with an RJ-11 jack. That kind of connection is common now but did not exist in 1960.
In the scene where Sinatra goes crazy upon learning that JFK will not be staying at his house, he begins pounding on a plaque which says "John F. Kennedy slept here November 6th and 7th 1960" JFK did not become president until January of 1961 yet we've already seen him in the White House and being referred to as Mr. President. Sinatra was even talking about building bungalows for the Secret Service which would not have been protecting him at that time. In reality JFK's visit to Sinatra's house was set for 1963 not 1960.
In the "Ocean's 11" filming scene, "Sinatra" can be seen wearing a two-button gray jacket. In reality, the original jacket was a three-button and was given by Frank to his son Frank, Jr., after the filming.