Continuity mistake: When Bruno and Bob are poring over exit polls, they mention that Santos seems to be leading in North Dakota, and comment that it is a state that hasn't gone Democratic in forty years. While that is true in the real world, in the fictional West Wing world, it is stated in Season 4 that Bartlet won the Dakotas in his landslide reelection.
Other mistake: In the flashback episode where the early days of the first Bartlet campaign are shown, Toby is "teaching" C.J. the proper technique for passing a basketball, something she is not very good at. Yet in this episode, when C.J.'s sexuality is questioned in the news, her history as a star basketball player in high school and college is revealed.
Plot hole: In Season 1, Episodes 5&6, it's mentioned several times that Zoey is already 19 a few weeks before she starts college. Yet in this episode, Charlie points out to Leo and the President later tells Oliver Babbish that Zoey was 17 when she was filling out her enrollment forms for college, therefore a parent had to sign them. These are enrollment forms, not application forms - it's extremely unlikely that they were filled out over a year before she started school.
Visible crew/equipment: At the end of the episode there is a camera visible in the oval office. After President Bartlet and Toby leave the office an attendant enters to turn off the lamps in the oval office. The scene widens as the lamps are turned off such that the audience can see the entire darkened room. The production equipment, a camera on tripod, and even a crewmember's leg are visible between the penultimate and final lamps being turned off. (00:41:50)
Factual error: In the scene where the Secret Service chief is in the car with Bartlet and he suddenly realizes Bartlet's injured, he yells to the driver to get to GW Hospital and the limo does a screeching U-turn on what appears to be the Arlington Memorial Bridge. At the time they were supposed to be heading from Rosslyn, VA, where the event was, back to the White House. But GW is in DC, not that far from the White House, so although the U-turn looks cool, it doesn't make geographical sense. Also, in that one shot of the outside of the car, it appears to be pouring, while back at the scene of the shooting as well as ahead of them at the hospital entrance, it's not.
Factual error: When Leo is reminding the Congressional Aides about their bosses' drug scandals, he states that Vicodin is a Schedule II Controlled Substance. At the time the episode was shot, Vicodin was a Schedule III Controlled Substance. Given his experience in this area he'd know better.
Plot hole: The Zoey Bartlett kidnap crisis cliffhanger that stretches across the end of Season 4 and the beginning of Season 5 makes no sense in terms of time. Charlie pulls out a note from his wallet that says they will dig up a bottle of champagne on May 7 - Zoey's graduation night. They do. Later that night she is abducted. She is gone three days. Four days later (in "Jefferson Lives") Abby chews out Leo, saying, "It's only been four days and her bruises have not yet begun to heal." The date should be May 14th. However, "Jefferson Lives" supposedly occurs on the Fourth of July - Zoey says it's the Fourth and they all watch the Capitol Fireworks at the end.
Factual error: Early in the episode, when Abby complains about who in their right mind would schedule outdoor inaugurations in January, Jed retorts by listing Jefferson, Adams, and Franklin, but inaugurations were in March until 1936 - not a mistake President Bartlett would make.
Factual error: Sam says the "state-of-the-art" oil tanker he recommends can carry 2.2 million gallons of oil and weighs 308,000 tons. The Exxon Valdez could carry over 60 million gallons (1.48 million barrels) and weighed only 211,500 tons. The writers mistakenly swapped 'gallons' for 'barrels' so that this enormous tanker carries very little oil. It would actually carry 2.2 million "barrels", which equals 92 million gallons.
Audio problem: As Charlie leaves the Oval Office, he pulls the door behind him. When it stops moving, the audio suggests it has been closed fully ("ka-chunk"), yet it's obvious it has not been closed as we can still see the outer office's curtains at the right of the screen, confirmed further when C.J. steps forward and the camera pans right to show the entire door frame area, and the door open quite a bit. (00:37:30)
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)
Continuity mistake: When Toby is walking back to his office from the press room with his ex-wife, the ex-wife's tag and chain keeps changing through the entire sequence. Sometimes it's over her jacket and the tag's just hanging or it's under the jacket and the tag is sitting just inside her jacket.
Plot hole: As Josh arrives at the security gate of the White House, protestors are gathered, chanting and otherwise talking loudly. He enters the gate, makes his way to the north entrance of the West Wing, and as he enters the lobby, the crowd is still heard quite plainly, just as loudly as at the gate, as if they were right outside the door, yet the protestors are much too far away to be heard so loudly, if at all. (00:00:40)