During outdoor scenes filmed for the 1999-2000 first season, the Washington Monument can be seen in the background encased in an elaborate blue mesh scaffolding used while the monument was undergoing a major renovation. However, in stock overhead shots used during the same episodes, the Washington Monument appears without the scaffolding (i.e., as it normally appears).
Several times characters refer to military hardware by incomplete names; for example: "AIM Sidewinder" missile instead of AIM-9 Sidewinder (as well as saying A-I-M rather than saying the word "aim"), "AH Apache" helicopter instead of AH-64 Apache.
First-season scenes shot on location in Washington D.C. show the Washington monument encased in the lighted scaffolding that was part of its renovation process. Aerial views of Washington shown in many of these same episodes (e.g., "Five Votes Down") do not show the scaffolding, indicating that these shots were taken from stock footage dating before the renovation.
Episode 4.02, Twenty Hours in America Part II, Donna, Josh an Toby are soaking wet when the go into the hotel, but in the next shot, when Toby an Josh have crossed the lobby they and their clothes are completely dry.
We see the inside of Josh's apartment several times throughout the series, but it's different every time we see it. The architectural style is the same, but the layout and the wall colors differ. We know that he hasn't moved because Donna notes in Season 7 that she has had a key for six years.
President Bartlet's code name, in one of the first episodes, is "Eagle." As he passes the Secret Service the agent says "Eagle has passed."
In a episode later that season when Bartlet has collapsed, an agent says "Liberty is down."
Secret Service code names for the First Family always begin with the same letter (for example, the Obama family code names are Renegade, Renaissance, Radiance, and Rosebud), which means that if President Bartlet's code name was Eagle, Zoey's would not have been Bookbag. Her code name would have begun with the letter E as well.
In several episodes they reference Myanmar. In fact, the US along with most western governments do not recognise the name change and still call it Burma. Various Government sites including the CIA Factbook confirm this.
Whenever President Barlett exits a limo, a secret service agent can be seen closing the door behind him. In actuality, the door is kept open to allow the president to re-enter quickly in case of an emergency.
At one point, National Security Advisor Nancy McNally makes a reference to "M2 Bradleys with 120 mm guns". In reality, M2 Bradleys are Infantry Fighting Vehicles and are armed with 25mm guns. This was most likely a mixup with the Main Battle Tank M1 Abrams, which is armed with a 120mm gun.