Factual error: In the scene where James is being told why Robert Carlisle's character is so deadly (no pain etc.), they mention a specific part of the brain - Medulla Oblongata, which the character has a bullet travelling through. This is part of the brain is responsible for control of breathing and the heart - you can't survive a bullet there...
Continuity mistake: In the scene of the boat chase, Bond smashes the machine gun holder, and some parts fall on to the Cigar Girl. When the camera goes back to the girl there are no parts on her.
Continuity mistake: In the first scene, when Bond has a gun to the Swiss banker's head, in the first shot Bond's got his gun pressed right up to the guy's neck. However, when it cuts, the gun is a couple of inches from the banker's face. (00:02:20)
Factual error: In the scene where Bond is watching the screen showing the bomb going through the pipeline, he asks for the distance the bomb has to travel and how fast it is going. Bond is told the bomb "is 106 miles from the terminal, going 70 mph." Bond says, "We've got 78 minutes." Wrong. At 70 mph (1.17 miles a minute) it would take about 91 minutes to cover that distance.
Continuity mistake: When M is imprisoned and is being spoken to by Robert Carlyle she is wearing a multi-coloured scarf which is draped over her left shoulder and down the left side of her body. When Carlyle leaves the room M turns around and walks towards the bed. The camera angle is reversed and she's not wearing the scarf. Later it is shown lying across the bed.
Factual error: When Valentin takes Bond and Christmas into the FSB's headquarters in Istanbul, he explains "This used to be the KGB's headquarters in Istanbul. Now we call it the FSB: Federal Security Bureau. Different name, same friendly service," or something to that effect. However, the important part that Valentin flubs is that "FSB" is the *Russian* acronym transliterated into the Roman alphabet. What it really stands for is Federalnaya Sluzhba Bezopasnosti. If he'd said the name correctly in English, he would've said "Federal Security Service," the FSB's official English name and that appears on all of its translated literature. Had a character unfamiliar with the FSB (Mr. Bullion, Electra, Bond, Christmas, and possibly even Renard) made this same flub, it would've been understandable. However, Valentin knows the FSB so well that he can apparently just waltz into a foreign headquarters building with whoever he wants. If he knows them that well, he's flat out not going to make so simple a mistake.
Continuity mistake: Despite running and jumping for her life, traveling at over 70mph, tumbling head over heels, and several massive explosions (during the nuclear factory and pipeline sequences), Dr. Jones' hair remains pinned up in a French twist with nary a strand out of place. Uh-huh. This is particularly glaring considering near the end of the film, the impact of an explosion finally does knock her hair loose.