The underwater city of BioShock is great and all, but is it something that could actually exist? Fortunately, we have these little things called "mathematics" and "physics" that allow us to answer such a question without, you know, going to the trouble of putting on a diving suit, hammer and nails in hand.
Play Noggin's Julian Huguet was fascinated enough by Rapture — and it's possible construction — to craft a seven-minute video on the topic.
Putting aside problems of getting down there with enough building supplies, the biggest problem would be pressure. Rapture is about 2000m below the surface of the ocean, which means it's under some 200 atmospheres. And before you ask, yes... that's a lot.
Too much, in fact, for Rapture to exist in the form it appears in BioShock. Way too much to have windows as thin and unreinforced as what you see in-game.
A couple more issue exist, but I'll leave it to Huguet to explain those.