Picture yourself as the lead designer in the development of a single player shooter.
From the get-go, you have one gigantic pitfall. You've said to your team (something along the lines of) "Right then! We've got the concept finished... But we need a campaign-y sort of thing, so - can anyone think of a story-line?"
First thing's first: Call of Duty, as with many other triple-A shooters, chooses to observe 'war stories' when faced with this same dilemma, and the remainder is the other half of the problem: they are in possession of a game mechanic with a primary focus on putting bullets in people. Real life would have the player cower in trenches for days and months in the wet and the freezing cold, but their mechanics aren't built to do that. It is for this reason that spectacle takes the fore, and we're left with exploding mansions, cliff-side chases, submarinal gunfights and nuclear war, all with plenty of enemies - the more things the player gets to shoot, the better use they are making of the mechanics.
The developers contrive to cram something vaguely resembling a plot into these bounds, and then expect the player to care when some team-mate dies, when all you've seen them doing - and all you've been doing - is killing everyone and exploding everything on the set-piece lovingly crafted by the designers. Any attempt to put heartfelt character development and thought-provoking dialogue into this world-wide catastrophe is rendered utterly dissonant with the essence of the game: hosing the enemy with death. It just doesn't work. Generally it doesn't get this far, you acknowledge; usually characters just turn into philosophers of bullshit in order to pass the down-time and plug up the plot-caverns.
Oh, but then there's Bioshock Infinite! This stimulates thought; it's one example of a game on the flip-side. It's a single player shooter, but with fantastic dialogue and multi-faceted character development, all set in a beautifully-crafted alternate universe. Guns may be involved, but the plot-thread is what's interesting.
The result of this approach received incredibly positive reviews upon immediate release, you remember, but once people had overcome their surprise ("A compelling shooter! Amazing!"), they began to concede flaws with the actual game-y bit of the game. The rust of neglect accumulated on the mechanics this time, and when it came to players doing the game-y bit (the bit that presents a challenge), the game felt in need of an oiling at the hinges.
Your line of thought travels thus, and the answer isn't (Y/N). A shooter can have an interesting story - the Bioshock series is proof-positive of that - but does that do the question any justice? When viewed in this light, you observe that a game's mechanics and its story seem to counteract each other. Trying to build a story around guns is a doomed endeavor; the result will feel contrived no matter what you do.
On the other hand, a game that focuses on story may as well be a film, because the decisions of the player and the shooting of the enemies take no part in the telling of the story. This presents a tangential thought: you could allow the player to choose where to go and who to shoot in a way which changes the course of the story, and thereby the game. But... Well, a game where the player is given choices that change the course of the story becomes exponentially more complicated; technology and time constraints mean that this route is also bound to failure (you think of Heavy Rain here; not a shooter, but an example of this ambitious game mechanic gone tedious and unpolished and ultimately boring). No truly viable options here, then.
Okay - hypothetically, the ideal in this quandary (as far as story is concerned) is to make the intrigue of the plot completely separate from any game-play. L.A. Noire manages this in bite-sized chunks, where cut-scenes develop the narrative and the negotiation of the challenge is for the player. Each mission is gripping, but not often because of the game-play. Meanwhile Assassins Creed: Black Flag does a fantastic job of making the plot interesting, thanks to some excellent script-writing, but you're forced to concede that these are both games where there is more to life than the death of enemies (surprising though it may seem for the latter).
So, you're back where you started. It's annoying to concede, but it seems that the shooter just doesn't have the elasticity to stretch as far as a story-line, without breaking itself in the process (or at least leaving it all straggly and useless). They should just... Stick to their job title. Forget about funneling a story into it.
Here you've reached the point of no return: shooter, movie or mess?