The game compensates for lag, and this compensation isn't always the best.
Whenever you see a ridiculous knife lunge, be it a second chance or standing, it's because the game interpreted your position to compensate for lag— whereas when you see it, you're far from the blade.
In conjunction with animations being "updated" for this effect, the result looks like a broken game. When actually, it's a poor netcode.
Very similar to this ( I couldn't find a BO example)...
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To put it simply, where you see another player is never where the player sees himself. They are usually ahead of where you see them. This also explains why you inexplicably can be shot around corners or running behind cover— you aren't always in the position you see yourself at because of the lag compensation!
Also, killcams are client based replays that are most often inaccurate. To get a "truer" perspective of an event, you would have to watch the killcam from the other player(s)' console.