Continuity mistake: When Doug, Toby, and Sam are talking to the President about how to handle the attempted repeal 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.
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. (00:10:05)