Swimming isn't easy to animate in a video game. First of all there's the complexity of underwater effects. Then there's the freedom of movement in three dimensions, which is neat for the player, but a real pain to achieve in third-person games in terms of achieving an intuitive interface and a fluid camera. Then there's the problem of water meeting land — it's easy enough to have your character jump or fall into water, but to get them out again you'd need ladders, shores, so forth, so you're going to have to do your art direction with that in mind. And to be even slightly realistic you'll need to think about the character's oxygen supplies. Basically, having your character be able to swim means adding an entirely new form of gameplay that's usually boring, annoying and nowhere near worth it.
The most common solution is to have no water, at least nothing more than ankle deep. But since our world is full of the stuff another very popular solution is to declare that the lead character of this game, no matter how athletic, trained and/or super-powered they may be, is just that: Lead. They cannot swim at all, not even enough to thrash around a bit. They just sink like a stone or are helplessly swept away by strong currents and are gone forever. It gives a new meaning to Hazardous Water.
Some more recent games have decided that it is worth it, and thus attempt more realistic solutions, with varying degrees of effectiveness. Quake, an early example, would deal damage to your health if you were underwater too long. However, armor protects against this damage — meaning you can stay underwater longer if you wear full body armor. Earlier still, the Ultima Underworld or The Elder Scrolls: Arena games didn't let you go underwater—swimming was essentially the same as walking except you bobbed up and down more and couldn't use a weapon, but after some amount of time your health would start taking a battering. More recently, the Tomb Raider games and the latter The Elder Scrolls have Oxygen Meters that would deplete as you swam, with running out causing lots of damage to be dealt, resulting in swift death. Other techniques have also been used to increase realism and decrease the prevalence of this trope.
Another solution is to put the main character in an environment suit, like in Halo. Which doesn't explain why the apparently perfectly ordinary land vehicles don't suffer from hydrolock when submerged. Even then, some bodies of water may still act as Bottomless Pits.
Even when the Player Character can swim, NPCs often still instantly drown, to avoid having to code swimming AI.
Interviews with developers note that this trope exists to avoid a few other tropes, such as Insurmountable Waist-Height Fence or Border Patrol as water is a more "logical" method than a fence or invisible wall. Additionally it removes the need for extra assets taking up inventory slots that don't get used a lot. Sometimes, an item that allows one to swim will be presented - usually to keep the player from going places they shouldn't be able to yet.
Important: If your drowning skills come with an Oxygen Meter, they're not Super. Only Super Drowning Skills go on the Super Drowning Skills page.
Compare Insurmountable Waist-Height Fence.
Contrast Super Not-Drowning Skills for characters who can stay underwater indefinitely with no apparent means of breathing, as well as Olympic Swimmer for those with more realistic submersion behavior but unrealistic swimming abilities.
Anyone who has eaten a Devil Fruit gains a special power, at the cost of losing the ability to swim in the ocean. In fact, having at least half of their body submerged will not only prevent them from using any abilities granted by the fruit, but will drain all strength out of them to the point of near paralyzation, causing them to sink. This is inconvenient, since most of the characters are sea-faring pirates on a planet with even less landmass than the Earth. Note that only actual submersion and seawater counts; rain is harmless and DF users can still take showers according to Word of God. A few people (mostly winners of the Superpower Lottery) have Devil Fruit powers that at least partially negate this weakness. Aokiji for example, An Ice Person , freezes the water just by coming into contact with it, giving him a massive advantage over other Devil Fruit users; also, Protagonist Monkey D. Luffy still has an elastic body regardless of whether he's in water or not, so he can have somebody else stretch his head outside of the water (he'll still be paralyzed, but he will then be able to breathe, at least).
Even before he ate the Devil Fruit, Luffy couldn't swim and was given the nickname "Anchor" by Shanks' crew.
The worst part: at times, Tony Tony Chopper would jump in after Luffy if the latter ended up in trouble in the sea, forgetting he can't swim either. Upon which Brook would jump in after Chopper, ALSO forgetting he can't swim.
Akane Tendō panics and cannot swim even when her life really does depend on it. Even wearing a flotation device, she manages to sink like a rock. Even when the water is only knee-deep. Until she was explicitly told otherwise near the end of the manga, she honestly believed you were supposed to breathe while underwater. Discovering that you have to hold your breath was nothing short of an epiphany for her.
Gentoku Ryuubi from Ikki Tousen. Usually played for laughs since she's a Dojikko as well. It's played more seriously in when she jumps into a huge pond to rescue a drowning Koumei while fully knowing the risks. They survive, but barely.
Misty's Psyduck doesn't know how to swim despite being a Water type. A few attempts have been made, but it ends in failure every time. He seems happier with an inner tube, though.
Even more fail on Psyduck's part given that real ducks naturally float.
...though maybe not so bad when you consider that a Psyduck, despite the name, is probably more of a platypus than a duck (no wings, doesn't learn a single "bird-related" move, can interbreed with mammals and assorted aquatic creatures rather than birds.) Unless platypi also naturally float, anyway.
Most Vampires in Vampire Hunter D cannot swim at all, and being submerged paralyzes them. And even D - a Dhampire of extraordinary caliber - can drown if he's in deep enough water to keep him immobile.
In Sword Art Online, Suguha / Leafa, in spite of being the most athletically gifted player in reality, can't swim, so during a trip to the pool with other players, Asuna volunteers to teach her to swim. While she gets a hang of it in the real world, she needs help from the others to get to an underwater dungeon in Alfheim Online.
In an episode of School Rumble Tenma (on behalf of her friends) challenges Nara and his buddies in a swimming competition. Since she knows she cannot swim herself, she expects that any of her friends will be able to win the competition (and the viewer may be led to believe that either Mikoto or Akira have better chances, at that). However, no sooner than they all dive, Tenma looks aside and sees... her three friends sinking behind her.
In the first episode, Sonic falls into a swimming pool and after trying to get out for several seconds, seems to give up and sit placidly on the pool floor, waiting to drown.
Even better, in the Sonic X episode "Sonic vs Knuckles", Sonic gets knocked into water by Knuckles. He instantly has a panic attack, splashing wildly and yelling. It takes him a few seconds to realise he's barely ankle deep, with Knuckles looking on at him with a wry smile.
Yoichi in Asu no Yoichi! jumps into the ocean to save a boy who fell in during episode 10. Unfortunately for him, he grew up on a mountain, so never needed to learn how to swim, and ends up having to get saved by Kagome, who everyone thought couldn't swim due to never participating in swim class.
In episode 4 of Yoake Mae Yori Ruri Iro Na, Feena jumps into a river to save a drowning puppy. However, since she grew up on the moon, and water is a scarce resource up there, she never learned how to swim, so Tatsuya ends up having to save her and the puppy shortly afterwards.
Hekikai No Aion: Seine can't swim. Even in a kid's pool with water-wings, she sinks like a rock. She also fights the people of the sea. The irony.
Masuzu in Ore no Kanojo to Osananajimi ga Shuraba Sugiru nearly drowns in episode 5 after climbing slowly into the swimming pool upon hearing rumors from other girls that she was deliberately not swimming to draw attention to herself from the guys. Eita is forced to perform CPR on her when he sees the swim teacher giving CPR to another student at the time, causing the other students to Squee at the "kiss" he's giving her.
Episode 2 of Angel Beats! saw a Dwindling Party scenario deep underground (it's the afterlife, so Death Is Cheap and Played for Laughs). Right before the commerical break, the group is down to four members. Fujimaki predicts newcomer Otonashi will be the next to die. Right after the break, the group find themselves in the aftermath of a water trap. Guess who's floating like a dead goldfish?
Episode 3 of Free!! The newest member of the ream, Ryugazaki Rei, reveals that he can't actually swim. He chooses to reveal this by belly-flopping into the pool and nearly drowning so Nagisa and Haruka have to rescue him.
Ledo does this in Suisei no Gargantia when he first tries out one of the Earthling's yunboroid. While his own mecha can indeed maneuver underwater fairly well, the primitive controls on the Earth yunboroid isn't intuitive to him, so he flails around a bit before sinking. Justified however, because he was born and grew up in space, so there was no need for him to learn swimming skills.
In Sonic the Comic this is played straight with Sonic then subverted. Handwaved by Sonic as him having taken swimming lessons.
Wolverine can't float and can barely swim because of the dense metal covering his skeleton. Same goes for Colossus in his steel form.
In chapter 3 of the The Legend of Zelda fic Twisted Fates, Link saves Medli from drowning in the sea after she was knocked out of the sky by Seahats. As Medli explains, the Rito are meant to be in the air, not the sea, and thus none of them can swim.
Also doubles as irony, since the Rito evolved from the Zora.
Stitch is so dense that he will automatically sink if submerged in water. Of course, he ends up on an island. Doesn't stop him entering Hawaiian Surf Competitions.
David in Unbreakable learns that along with his superhero-like strength, he has an Achilles' Heel of water. Justified in that David both has a fear of water due to a childhood incident, and in the context of the movie, his powers are ill-defined. David is vastly healthier than normal (never getting sick, and being vastly stronger than his size indicates), but no limits are recorded (he stops measuring his strength before finding his limit, and while he is the sole survivor of a massive train wreck, he balks at trying to see if he's bulletproof). Since humans can't exactly survive underwater, either, his "weakness" may simply be panicking from his childhood combined with still needing oxygen.
It's justified even further at the climax of the film. When David is shoved into an above-ground swimming pool he gets caught in a tarp. Even a person without his weakness would be unlikely to survive that.
In Superstar, Slater can't swim, and still doesn't bother to learn after he almost drowns and Mary Katherine saves him the first time.
After being knocked off his boat, the killer in Terror at Tenkiller quickly drowns, despite being a marina worker. The Final Girl has no trouble swimming to shore.
The famous 3-breasted cat-woman in Star Trek V: The Final Frontier. She leaps onto Kirk's back, is airplane-spun for a half-turn, then just gently tossed face-down onto a table where a variation of billiards is played, with a quarter-inch of water covering the playing field. She then just lays there, presumably insta-drowned.
Garion had acquired this skill ever since he was hit by a log on a pond while attempting to swim.
In Fairy Dust And The Quest For The Egg by Gail Carson Levine, it is mentioned that if a fairy's wings got wet, they'd become heavy and drag the fairy underwater.
Dwarves in the Dragonlance series are noted for swimming like stones.
It becomes a running joke in Aubrey-Maturin that Doctor Stephen Maturin would always nearly drown when trying to board a ship. Considering how often he needs to go on and off ships as the Naval Surgeon...
In the Green Sky Trilogy, Raamo slipped and fell into the Bottomless Lake and made no effort to try and swim or even let go of the "tool of violence" he was carrying. Snyder, realizing she really screwed up with this ending, authorized what was likely the first canonical video-game sequel to another medium's story...and even in that game, so much as touching the water would send the player back to their home with an ominous "You were found unconscious" message.
Septimus near-drowns surprisingly quickly in after he's been thrown into the water by Queen Etheldredda.
In the end of the book, Etheldredda drowns very quickly as well after she's slid into the river.
In the Guardians of the Flame series, dwarves are noticeably denser than humans, with the result that they sink like a stone in water. This leads to a panic-inducing incident in one book when the dwarven member of the cast is tossed off a ship in the middle of a battle, with no one available to rescue him. Fortunately the ship is anchored, and he manages to get to the anchor chain and climb up it before drowning.
In Edgar Rice Burroughs Martian books, very few Martians know how to swim, because Mars has very few deep bodies of water.
Live Action TV
In Smallville, Lana manages to drown in less than 15 seconds. Don't worry, Clark saves her with some mouth-to-mouth.
In LOST, Sayid manages to be non-resuscitatably drowned about 5 seconds after he stops struggling.
Parodied in "Glass of Water", where a bunch of klaymen drown in a tiny puddle produced by a glass of water by sticking their faces in the puddle. And then several more are drowned by the towel used to clean up the puddle.
In Code Lyoko, any monsters or heroes falling into the Digital Sea are "permanently virtualized", i.e. dead. Happened to a lot of monsters, and to Yumi and Aelita once each (they got better). There are also ordinary bodies of water in some sectors; terrestrial monsters just sink and drown there, but the heroes can swim through them.
SpongeBob SquarePants: Neither Spongebob nor Patrick know how to swim. Ironickly, they both live underwater.
Some accounts of Fredrick I, the Holy Roman Emperor actually have him falling face-down in plate armour into a shallow part of a river and, you guessed it, drowning. Justified in that plate armour is not recommended swim-wear, it's extremely heavy, though it's a myth that armored knights could barely move around in it as it's well distributed. That said, falling off a horse is dangerous, moreso when you've got an extra 60 pounds on you. There are myths of knights drowning in puddles, but if that happened, they were likely concussed first. The whole incident may also be a simple myth that never actually happened.
The Battle of Lake Peipus in Lithuania in 1242 saw this trope played out on the grand scale. Prince Alexander Nevsky's lightly-armed Russian soldiers drew an army of impetuous German knights out onto a frozen lake where they were at a disadvantage. Not only did the western crusaders find it hard to keep their footing, the Russians lured them to a section of the lake with thinner ice that could not take the weight of Teutonic Knights in full armour.
In at least one episode in WW2, light Russian tanks crossed frozen rivers, daring German pursuers to do the same. But a Tiger tank weighed in at fifty tons as opposed to twenty-two...
Modern soldiers are known to drown because of the encumbrance by their combat gear. Many soldiers and marines landing on Normandy or various Pacific islands during World War 2 were killed for this reason. Even in 21st century, training for operations in water while wearing full military gear is considered among the toughest exercise for soldiers.
This applies to pretty much all of the great ape species aside from humans. In addition to a more tall, thin body design (granting a more efficient stroke), humans have a higher percentage of body fat and larger lungs in proportion to mass than other large primates, giving them greater buoyancy. Still, humans are poorly equipped for swimming in comparison to many mammals, and very skinny humans often have trouble swimming efficiently because their legs and feet tend to sink.
It should also be noted that humans are the only ones that have small swimming webs at their hands and feet. Apes do not have these. This is another hint that swimming must have been at some point important to our species.
Another interesting detail: babies can swim the moment they leave the womb. They can even take the longest dive any human can ever do, because they are still connected to the placenta and thus do not need to breathe (just yet, anyway) for about 10-20 minutes, until the placenta stops working. Unless you cut the umbilical cord earlier to unnecessarily make birth a lot harder for the newborn.
That's actually a serious misunderstanding that's been passed around as truth. While there are some bits of truth to the idea that newborns can swim, it's really not the case at all: 
Before actual swimming techniques were developed, a lot of people could easily commit suicide by wading into a lake until the water level went over their heads.
The aforementioned Bolton Strid. It may look like a gentle brook, but it holds the amount of water one would expect from an enormous, raging river. How? The river's really deep, and the flow's similarly intense below the surface. The result? It has a 100% lethality rate. Yes, that means falling inside is sure to kill you. You'll be lucky if the body even shows up downstream.