Once you get down there, it’s all broken and filled with water in places, and the Splicers are wandering around as if they’ve been mad for quite some time. How long? Years?
According to the storyline, between two and three years. The game takes place sometime in 1960. Atlas’ attack on Ryan, which is what wrecked the city, was NYE 1958.
(Politically, it had been going downhill for a longer time than that. But the war is what made it look flooded and broken.)
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You could read their wiki or the novelization.
http://www.amazon.com/BioShock-Rapture-John-Shirley/dp/0765367351