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It varies by the type of submarine. United States, Russia, United Kingdom, China, France, and India are examples of nations who have or are assumed to have nuclear submarine capabilities.
For security purposes, governments don't give out full specifications on their weapons of war, especially on something as big, expensive, and instrumental as a submarine. The true depth information is classified, probably above "Top Secret". For the general US submarine fleet, the "official" answer is typically around 600 feet. (According to one Navy instructor it was changed from 400 feet when a documentary was aired accidentally showing footage of the depth meter at a depth of around that depth.)
Any Navy in the world who operates a nuclear submarine is going to publish a low ballpark figure in the name of national security and secrecy. If everyone knew the exact depth a submarine could submerse to, it'd be easier to kill them. Wikipedia says a Soviet/Russian class Alpha submarine may have successfully operated at a depth of 4,300 feet. But rest assured whatever the published depth of submarine is, it many times greater than that.
