Corrected entry: There are occasional references throughout the show to the Bartlet family owning a farm in Manchester, NH. Manchester is the largest city in that state, and not in a rural area: saying they "have a farm in Manchester" would be like saying they "have a farm in Baltimore" (or Glasgow, for the UK readers among us). No one with roots in New Hampshire as deep as the fictional Pres. Bartlet's are supposed to be would ever be so inaccurate as to refer to the agricultural parts of the state as being Manchester.
Corrected entry: Political consultant Mandy Hampton disappeared completely with absolutely no explanation between the first and second seasons. She was in the last episode of the first season and not in the first episode of the second (which took place only a few minutes later) and no one ever explained where she went.
Correction: Her departure is not explicitly explained to the viewer, but the fact that the show's characters are not similarly confused suggests they are aware of when and why she left, and have moved on. The actress' departure as well as the choice to not specifically explain it 'in-show' were both intentional, and are not mistakes. (Humorously, the writers could easily have killed her with the season-ending shooting, nice and smooth)
Corrected entry: In the first season, every employee in the west wing (except the president) is wearing an ID badge. Since then, only a few people are seen wearing them, but surely the policy of the White House hasn't changed since then?
Correction: Many people that live in the outskirts or suburbs of a large city say that they live in the actual city so that people not from that area know where they live. They may have meant the outskirts, or just outside of Manchester, rather than directly in the middle of it.