Continuity mistake: When Josh is walking down the White House hallway in the office area, he is talking to someone, and at first there is nothing in his shirt pocket, but as the scene progresses, we can see his glasses in the pocket. He did not put them there during the scene.
Character mistake: During the face-off about religion/politics, they get into a shouting match about commandments, both groups being equally convinced that "honour your father and mother" is the 1st or 3rd commandment, respectively. In actual fact it's the 4th or 5th, depending on which version (Catholic or Protestant) of the 10 commandments you go by. See http://www.positiveatheism.org/crt/whichcom.htm.
Factual error: A secret service agent goes to Leo McGarry and tells him Sharif has crossed the border from Canada into the US - he says "from Ontario into Vermont". It is about 50 miles from Ontario to the Vermont border - the only province that borders on Vermont is Quebec. At this high level of intelligence, this could not be a character error - just the West Wing writers not checking a map.
Factual error: The fighter jet alongside Air Force One has "NY" tail markings which designate it's with the 174th fighter wing, with a home base of Syracuse, New York. Surely, in this emergency over Washington, DC, jets from a more local base (like the 113th's F-16s at Andrews AFB) would have been dispatched to aid the President's plane. (00:13:05)
Factual error: When the British Ambassador, Lord Marbury, greets Abbey at her party, he tenders best wishes from "Her Royal Majesty", his Monarch. However, as Her official representative, Lord John should have been cognizant that the reigning British monarch is referred to strictly as "His/Her Majesty".
Continuity mistake: Near the beginning of the show, Josh is talking about baseball while looking at some stapled political papers - the top one is folded back over the second one. As Donna asks "what is it?", the first page is falling free and Josh is only holding the second one, but he had no time to adjust them.
Factual error: Hal Holbrook's character is called in to assist in solving an incident where a US sub has gone missing in hostile waters. Despite being elderly, he is considered an expert in such matters, and is advising the President in that capacity. Yet while relating similar historical submarine incidents, he makes two significant errors. First, he describes the "Glomar Explorer" and the K-129 Russian sub as "two subs", but the Glomar was a surface ship, not a sub (See http://www.espionageinfo.com/images/eeis_02_img0482.jpg). Then he states the USS Gudgeon was trapped by Russian ships for four days, but the entire incident took 30 hours. He is never challenged on these facts, and his advice is unquestioned. Rather than character mistakes, as the character's meant to be an expert, this is more likely bad research and embellishment by the writers. (00:28:00)
Continuity mistake: Bruno and C.J. sit down in a meeting and are talking about women voters. When the shot is on Bruno, we can see C.J. is holding a black folder that is partially open. When the shot goes to her, though, she has her hands on the folder, which is opened on the table.
Continuity mistake: Newly hired Assistant White House Counsel Joe Quincy is looking over his new "office". He's told it is the office traditionally given to newly hired lawyers, and is known as the "Steampipe Trunk Distribution Venue". In prior episodes, Ainsley Hayes, the previous Assistant White House Counsel, was also given this office, but this space is significantly altered in size from the earlier appearances. Essentially, the length of the space has been cut in half and the intervening wall with door is gone. Yet the room is not newly remodeled. It's a basement space with old pipes and walls, etc. The makers simply shrank the set without explanation. (00:06:30)
Continuity mistake: Early in the show, Toby, Sam, and C.J. are at a conference table, talking about a concession speech. C.J. is holding a mug and scratching her hand, but as she turns and says "ooh, Mr. Lyman", her mug is suddenly sitting on the table. In the next shot, as she is talking to Josh, her hand is around the mug. None of these changes happen in natural time.
Continuity mistake: During the entire first half of the episode President Bartlet is using a cane and is walking with a very pronounced limp due the effects of his MS. However after changing from street clothes into a suit to meet the President Elect he is walking fine with no symptoms whatsoever.
Factual error: When Sam is telling Leo about the couple who crashed their car after the fundraiser where the President talked about seat belts, he states, "Now she's suing him for contributory negligence!" Contributory Negligence is not something you can sue for, it is a defense to a lawsuit and is based on the plaintiff's own negligent conduct - the woman would be suing the president for just "negligence." Sam repeats the phrase at 7:45, when he claims "Contributory negligence in wrongful death is the tort equivalent of murder." This makes no sense from a legal perspective. As an attorney, Sam would know the difference - this a writing mistake. (00:01:52)
Factual error: When President Bartlet is on the phone with the butterball hotline, he claims to be from Fargo, ND. Charlie tells him the zipcode of Fargo is 50504. Fargo has 13 zip codes assigned to it. None of them are 50504. They are all beteeen 58102 and 58126. (00:28:05)
Continuity mistake: When Leo asks Chairman of the Joint Chiefs Admiral Fitzwallace about the optics of hiring Charlie, a black man, to carry the president's bags, Fitzwallace's ribbons on his Navy uniform are upside-down. This mistake is very visible because he has two ribbons on the bottom when there should always be three. On his way out the door, there is one shot where Fitzwallace's ribbons have been fixed, but then as he walks out the door his ribbons are upside-down again. (00:25:50 - 00:27:10)