Continuity mistake: Josh and Amy are in a restaurant having breakfast and talking about the vote - the sugar dispenser can be seen next to some yellow flowers. In subsequent shots the flowers are still there but the dispenser has disappeared, even though no one touched it.
Audio problem: Toby enters the Oval Office at the end of a meeting the President is holding with about a half-dozen people. The meeting breaks up and Toby follows Bartlet toward the desk as the others meander toward the door. A few seconds after the camera pans away from the others, we hear the door close and the room is now silent as Toby and Bartlet begin to converse. But the door closing happens way too soon for all those people to have gotten out of the room. Two men are especially just standing there, looking as if they aren't planning to leave at all. The other doors aren't used nor heard.
Continuity mistake: When Doug, Toby, and Sam are talking to the President about how to handle the attempted appeal of the estate tax, they mention several times that it will be the President's first veto, and that it is something that he has never done before. However, in episode four of season two, "In this White House", in the scene where Sam is debating Ainsley Hayes on Capital Beat, the moderator asks why the President's education bill is different from the Republican proposed bill "which the President vetoed." So, the veto of the estate tax repeal was not the president's first.
Continuity mistake: Josh is getting coffee and talking to another guy about a Shakespeare production Bartlet is supposed to attend. When he goes to the coffee machine, his backpack is over his shoulder and the strap is near his collar but not covering it. When he walks away from the coffee machine, most of his collar is covered and quite dishevelled from the backpack strap. At no time did Josh adjust the backpack.
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.
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: 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".
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.
Plot hole: Joey and Kenny are led to the Oval Office by Charlie through the Presidential Secretary's (and his own) office. Charlie ends the scene saying, "Okay, you're in the Oval Office" as he sends Joey and Kenny that way. Charlie always knows the President's whereabouts and schedule. Yet moments later, Josh arrives in Leo's office where others have gathered to await the start of the same meeting - and Joey and Kenny are also here. Then everyone goes into the Oval Office to await the arrival of the President. It is as if the makers forgot Charlie led Joey and Kenny here moments earlier.
Character mistake: Jed Bartlett is talking to Leo and Hoynes. He says 'Calvary' instead of 'Cavalry' in the line "Tell Josh that General Pulaski was a Polish Brigadier General who vanquished the Russian and Prussian military then came to the colonies and commanded our cavalry during the American revolution."
Factual error: Lord John Marbury says he is "the Earl of Croy, Marquess of Needham and Dolby and Baronet of Brycey." His title according to the British peerage rules would be "The Earl of Croy" and he would be called Lord Croy in conversation. Lord John Marbury would be what one of his younger sons also named John would be called.
Continuity mistake: When Josh is drinking water from a bottle he starts screwing the lid back on as he moves it down towards his desk. In the very next shot, however, not only is the bottle suddenly on the desk without Josh's hands anywhere near it, it also doesn't have its lid on.