One of the more notable "near misses" in the US Navy submarine service occured in 1969. The USS Chopper, at that point a 25 year old Balao fleet boat converted to Guppy Ia, had a mishap that stood the submarine almost vertical, bow down, in the water. Before control was regained, the bow was estimated to have reached a depth of 1100 ft, and the stern in excess of 700 feet. During the recovery by the crew, the bow and stern quickly swapped places and the submarine rose almost vertically, now in bow up configuration. Test depth for this class was roughly 420 ft. (when built).
A blow by blow account of the incident is here: http://www.usschopper.com/Chopper%20Deep%20Dive%20Report.htm