The greatest depth in the ocean is the Challenger Deep in the Marianas Trench off Guam. At about 36,000 feet, it has altready been visited by a submarine of special design ("Trieste" in 1960) that carried its two crewmembers in a titanium sphere. The interior was not pressurized.
To increase the operational depth of a submarine, it would be possible to pressurize the interior to balance the outside sea pressure, but in practice, it is more practical [I suspect] to construct a smaller crew compartment and build a stronger hull. [Not being a marine engineer, I can not speak in depth to submarine design.]
Problems associated with pressurization of the interior are many, some of which would be:
[1.] Nitrogen is narcotic as the pressure is increased (nitrogen narcosis), so it would be necessary to compress with helium. Helium is however, notorious for being able to penetrate glass and metal with relative ease; every thing in the submarine that is electrical and would be bothered by helium and would need to be somehow protected. Anything with a vacuum inside would be particularly susceptible (e.g., cathode ray tubes and television tubes could not be used).
[2.] Pressure is not without its effects on living systems. Pressure changes the conformation (= shape) of molecules, and it can produce effects of its own some detrimental. Thus large pressures on the interior could probably not be used.
[sp]Pigs have successfully gone to 5,000 in a chamber and been returned, so we know that this is physiologically possible. How this affected the mental status of the pigs is not known.
[3.] The higher the interior gas pressure, the greater the gas saturation of the crew, and the longer it would take to exit the submarines interior. If the sub were compressed with helium, a rule of thumb is one day per 100 feet of depth decompression time.
[4.] Everything connected with the climate control system would need to be redesigned. The interior gas would be very dense. Sound would travel better in this gas and might present a problem regarding passive detection of the sub by another enemy submarine.
Final comment is that, yes, it would be possible to compress to a certain point and then decompress and get the crew out. The deepest depth to date in a chamber run with humans, that I can remember, is a bit more than 2,000 feet. Decompression required three weeks.