I'm surprised this is (judging by the reactions in this thread) such a controversial and argued issue.
It is for the simple reason that most games have really poorly implimented lag compensation, as in, it's barely limited if at all.
Lag comp works quite well at low to moderate pings, it lets you aim at where you see the enemy and shoot, and the guy will be hit very shortly thereafter, at low pings allmost instantly, at moderate ones he might take a few steps.
But at high pings, lag comp becomes an annoyance to everyone around you, where you can run 50 meters after having been shot and then suddenly drop dead from seemingly nothing.
Just one guy on the server with an atrocious ping can really cause confusion, and massively break immersion.
That's not lag comp's fault though, when used well, with reasonable limits, it can make the game playable at slightly higher pings than without any lag comp, but it should disable when you get above a certain ping, so it's only the player with the horrible ping that gets annoyed by his own bad connection, and not everyone else on the server (yeah there really is no substitute for a good low ping server, if you play with a massive ping, then either you or someone else will have to deal with that, and that's just how it is).
But when applied too liberally, lag comp becomes really annoying, as it basically makes other peoples bad connections your problem.
Sadly, it seems most Dev's do apply it to liberally, probably because they don't want to see reviews about how "this game lagged a whole bunch and i coulden't shoot anything! it was terrible!", so they set the cutoff limit much to high so everyone could basically get kills despite horrible pings.
So bottom line: Used well there is nothing wrong with the system, and it would actually be a good thing, but most people have really bad experiances with it, because it's rarely used well, and so they rage whenever it's even mentioned.