Deliberate mistake: This show is famous / infamous for having many long conversations between characters that are walking the corridors of the West Wing of the White House. In this episode, the chat was long enough that more space was needed to complete the walk, so the actors were instantaneously teleported to a different spot to lengthen the walk. Using http://www.whitehousemuseum.org/special/wwtv.htm as a guide, Pres. Bartlet and Leo leave the Oval Office, pass through the Presidential Secretary's office, turn left, passing between the Roosevelt Room and the Chief of Staff's office. At the end of the Roosevelt room, they turn left through dark wooden doors that are open. Instantly, the actors are at the next junction higher on the map (no wood doors), as if they'd just passed between the Roosevelt room and the Communications Bullpen, and they continue (downward on the map) through the intersection where they original turned (watch for the doors after Leo says "You wouldn't understand"). (00:02:40)
Deliberate mistake: When James is shot on the courthouse steps, the crowd of reporters that has gathered quickly scatters to escape the gunfire. One of them is a woman in a black windbreaker holding a camera. The camera cuts away briefly and then back to the crowd - you see the exact same people, including the woman in black running in the exact opposite direction as before. The footage was apparently mirrored and repeated for the second shot of the fleeing crowd.
Deliberate mistake: Tony smashes AJ's front windshield of his Nissan SUV one time with a plastic sports helmet and it leaves a hole 4 times the size. Not only is the scale size of helmet to hole incorrect, but front vehicle windshields don't shatter like that. Modern windshields are made of laminated safety glass, which consists of two curved sheets of glass with a plastic layer laminated between them for safety, and bonded into the window frame. It would've taken many impacts to cause such a hole, not one hit. (00:49:00 - 00:49:45)