Character mistake: When Bond is jumping from the dam in the opening sequence, you see him 'clip in' twice, once attaching the rope to his feet, and once attaching the rope to the dam. He is using screw-gate carabiners, large clips that have a small bevel gate on them that secure them shut. Yet Bond doesn't screw them shut - all it takes is a simple flick of the fingers and they're done. This means that they're not as safe as they should be, and could potentially open - particularly when doing something as stressful as diving off a dam. You would have thought that as Bond's parents 'had the luxury of dying in a climbing accident' as Trevelyan puts it, he would take more care over basic safety procedures when using ropes and climbing equipment. (00:01:10)
Character mistake: At MI6, when M is showing Bond a profile of Ourumov on a screen, Arkhangelsk is misspelt as "Arkhangeesk." (00:46:36)
Character mistake: When Boris teases Natalya about not telling her the password to his spike program, she glances down at the keyboard as he types the mystery password ("chair") and should've been able to make it out, given her level of programming expertise. (00:30:26)
Chosen answer: The real answer, which I believe was explained better in the game or in special features. Regardless, I did hear, is that Ourumov wasn't aiming directly at Alec's head, but to the side and shot the ground behind him. But made it look like he shot him in the head. Alec would have felt the bullet whiz by him.
That makes more sense.
It has been a while since I watched the movie, and my copy is a VHS... but I believe if you watch very closely you can actually see the bullet hit the ground behind him with a tile breaking. I don't remember for sure if you can, but I seem to remember they did add that little effect.