When Sydney first comes to the White House as a representative of the GDC, she tells the guard at the gate this is her first time to the WH. Later, she tells Shepherd that she'd been on a tour of the WH before.
When Lucy takes the book it is rotated 180 degrees from the position it was when the president hands it to her.
During the Virginia Ham scene, the clock behind Sydney's boss changes times between shots.
When President Shepherd is going to Lucy's room after making the deal with the Motown Three, he unbuttons his suit jacket in the hallway. When he enters the room and she informs him that Sydney is upset, his jacket is buttoned.
Near the end of the movie, when President Shepherd is speaking with A.J. and putting on his coat, at one point in the scene (while the camera is focused on A.J.) he is holding his scarf in his hand. The very next camera shot, the scarf is neatly arranged around his neck and tucked into the front of his coat.
At the end of the movie, when The President is getting ready to leave and drive to Sydney's house to beg for her back, she walks in and declares that the traffic is awful. This doesn't make sense; she wouldn't have been able to hear his conversation through the wall because the Oval Office is soundproof.
Just after inviting Sydney to a dinner, the President is seen entering and taking off in a helicopter. As a security measure, Marine One always flies in ever-shifting groups with identical decoy helicopters, sometimes as many as five. However, in the film, the helicopter escorting the President is alone. There is a repeat occurrence at Camp David later in the film.
The time frame of this movie is approximately from the end of November thru the beginning of January. During that time, Senator Bob Rumson would've been in the middle of the Republican Primary, and that would be what his time and effort was used on winning, not the presidency. He more than likely would not start a full campaign against the President until the primary was over, and he was declared the Republican nominee for president.
After the President's big speech at the end of the movie, staff members are walking quickly back to re-write the State of the Union address. After Leon says "Well, you don't see that every day," Louis says "Yeah - he's got the members of the press corps asking each other how to spell 'erudite.'"
Although the president never uses the word 'erudite' in his speech, erudite is an apt description of his demeanor during the speech. This was a reference to his demeanor, not a reference to something he said.
While the US President can't introduce a bill in Congress, it's common practice for him to get a sympathetic legislator to do it for him; this could be colloquially referred to as the President introducing the bill. In any case, he says, accurately, that he's "sending a bill to Congress for its consideration" which is perfectly proper as part of the State of the Union message.
Near the end of the movie when Sydney asks how he was able to send flowers to a woman while he was President and The President says "It turns out I have a rose garden." Since the State of the Union address occurs in January it is highly unlikely for the rose garden to have been in bloom in January in Washington DC. However, given some of their previous joking interactions, it's likely the President was just being a smart-ass.
After Shepherd's press conference, Lewis says everyone is looking up how to spell "eriudite." Lewis would know the word is spelled with only ONE "i," but it's likely he was only joking and being sarcastic hence the laughter from AJ and others.
In the scene where Sydney and AJ meet to discuss the legislation regarding the reduction in fossil fuels, behind AJ is hung the famous Trumbull painting of the Committee of Five presenting the completed Declaration of Independence to John Hancock and the Second Continental Congress (the subject is sometimes erroneously believed to be the document's signing). John Hancock is seated to the left of the viewer instead of the right, indicating, that the entire scene has been flipped.
At the State Dinner with the French President, when people are shown entering the White House and passing through the metal detector, the detector is clearly not switched on. They are also moving at a reasonably fast pace, sometimes passing through at least two at a time.
Standing clock in the Oval Office shows randomly varying times. When the President asks for the time and someone answers "3:30," the clock reads something completely different.
When the President drives in the limousine with Robin and Louis, in almost all the shots before the Limousine pulls out of the White House gate, the background visible through the car windows doesn't move smoothly but rather jittery, at a slower frame rate. The background projection is obviously slowed down to simulate the slow driving inside the White House premises.
When the President orders a missile strike in the middle East, he's in a meeting in the evening in the White House. When he asks which shift is on duty at the target, they say the night shift. Evening in DC would be midday in the middle East.
In the scene where the President first calls Sydney (at her sister's apartment) to ask her out, the whole time of the phone call you can see the crew on the window reflection
Sydney gets votes for the Greenhouse emissions bill by promising "full White House support," even as the President is losing votes on his crime bill because the President's popularity is plummeting. She should be losing votes the same way the President is.
At the state dinner, it is suggested that the French president and his wife are being left out of the conversation because no one at the table speaks French. But when Sydney saves the day by speaking French to the couple, the French president follows up with a few sentences in perfect English, which his wife clearly also understands. The whole sequence makes no sense.
During a White House Christmas party, Sydney recounts an incident on Dupont Circle, prompting AJ to ask "What were you doing up on the Hill." But Dupont Circle is northwest of the White House and Capitol Hill is to the southeast. She was away from the Hill.