You can swim to any depth, but it might kill you
The Darkened Depths: Elemental Plane of Water
DMG, pg 56
The deeper extents of the plane, where no sunlight reaches, are called the Darkened Depths. Horrid creatures dwell here, and the absolute cold and crushing pressure mean a swift end to creatures accustomed to the surface or the Sea of Light. Krakens and other mighty leviathans claim this realm.
So it seems that it means a swift death for anyone who is accustomed to breathing above the surface or even in shallower waters, once you get to some depth below the water. This effect is explicitly nonmagical because the reasons for death are cited as "the absolute cold and crushing pressure."
There is no depth listed anywhere for where the Sea of Light gives way to the Darkened Depths, though. This is probably intentional, but if we use real world sunlight as our guide. Sunlight does not reach depths of below 1,000 meters (3,280 feet).
Also note that there are rules for dealing with Extreme Cold (if you consider this depth to be that cold, which it seems to be given the description):
DMG, pg 110, Extreme Cold
Whenever the temperature is at or below 0 degrees Fahrenheit, a creature exposed to the cold must succeed on a DC 10 Constitution saving throw at the end of each hour or gain one level of exhaustion. Creatures with resistance or immunity to cold damage automatically succeed on the saving throw, as do creatures wearing cold weather gear (thick coats, gloves, and the like) and creatures naturally adapted to cold climates
DMG, pg 116-117
Swimming through deep water is similar to traveling at high altitudes, because of the water's pressure and cold temperature. For a creature without a swimming
speed, each hour spent swimming at a depth greater than 100 feet counts as 2 hours for the purpose of determining exhaustion. Swimming for an hour at a depth greater than 200 feet counts as 4 hours.
So once your PC reaches a depth of 100 feet, every hour of travel counts as 2 hours for the purposes of determining exhaustion (note: you can travel for 8 hours per day without rolling for it), limiting you to 4 hours of actual travel time.
Once they reach a depth of 200 feet, swimming one hour in it counts as 4 hours, so you can only travel for 2 real hours before you have to start rolling.
Most PCs without a swim speed will typically have a speed of 30 ft, so they can swim 15 ft in 6 seconds. They can therefore reach a depth of 200 ft in 80 seconds, so this depth can be reached before you have to start making saves.
Additionally, assuming the same above (no swim speed, 30 ft walk speed), a PC can reach a depth of 3280 ft in less than 22 minutes, where they will presumably enter the Darkened Depths if they were in the Elemental Plane of Water. At this depth, they may very well die.
But the full extent of swimming at deep waters is... deeper
However, on top of the penalties to swimming in the deep, you have to also roll per hour of swimming. So if you were at a depth below 200 ft and swam for one hour, you are considered to have been swimming for 4 hours for the purposes of determining exhaustion. But you still have to roll a DC 10 Con save or gain one level of exhaustion:
DMG, pg 116
Unless aided by magic, a character can't swim for a full 8 hours per day. After each hour of swimming, a character must succeed on a DC 10 Constitution saving throw or gain one level of exhaustion.
This means, if you swim for two hours, you have to roll two DC 10 Constitution saves, and you are considered to have been traveling for 8 hours. If you swim for 3 hours under depths of 200 ft, then you must roll twice more:
One DC 10 Constitution saving throw for the extra hour of travel
One DC 14 Constitution saving throw for traveling the equivalent of 12 hours under the Forced March rules (DC = 10 + 1 per hour traveling above 8 hours).
And for every hour of swimming beyond that, you have to keep rolling twice. The first roll has the same DC throughout, while the second roll's DC increases by 4 each time.
The rules are the same for swimming under 100 ft, but the DCs eventually play out the same. You just take longer to get there.
Note: This is before taking into account the rules for Extreme Cold, if you were swimming in waters at or below 0 Farenheit.
None of these penalties are in effect for creatures with a swim speed
With the exception of the Darkened Depths, creatures with a swim speed do not suffer from these penalties.
DMG, pg 116
A creature that has a swimming speed -- including a character with a ring of swimming or similar magic -- can swim all day without penalty and uses the normal
forced march rules in the Player's Handbook.