Continuity mistake: When the Presidents jump off the moving train, James Garner pushes Jack Lemmon off first, waits several seconds, then jumps himself. Given the speed and direction the train was going, Garner should have landed quite a ways away, to the left of Lemmon. But he ends up only a short distance away, to the right of Lemmon (based on the direction they were facing when they jumped.)
Continuity mistake: When hillbilly family are driving the Presidents around, in the scene where they are arguing about the country, there is a field of plants on Garner's side. When it cuts to Lemmon, there is a rock wall, but when it cuts to the woman in front they are suddenly driving beside a gas station, where they stop to throw the garbage.
Character mistake: When discussing the potential fallout of ousting the new president, Douglas says that because there's no vice president that according to the 25th Amendment the Speaker of the House is next in line. Not true. The Amendment itself is silent on the order of succession after the Vice President. The Constitution leaves the order of succession after that up to Congress, and the Speaker of the House is listed according to the Succession Act of 1947, not the 25th Amendment.