In many works that revolve around martial arts, magical skills, or other powers that can be passed along by training, those arts or skills are portrayed as vastly powerful. If any main characters gained these talents and powers, they'd be almost unstoppable; alternatively, there's a special skill or ability they need to learn in order to win the final battle or achieve a central goal. But instead of completing their training ahead of embarking on a quest or fighting an enemy, something causes them to head out before their training is finished. Perhaps the villain attacked early, maybe the mentor possessing the skill is incapacitated before the training can finish, or maybe the character is simply too arrogant or undisciplined to finish their training. Whatever the reason, the character lacks a power or talent they should have.
Inevitably, the fact that this training was incomplete will become a plot point. Sometimes, especially when they're The Hero, the character develops this power on their own when they really, truly need it. Alternately, many heroes turn out to be naturals at the whole thing and don't really need any training, or they make a virtue of their indiscipline by adopting a form of Confusion Fu. Compared to people fully trained in the same technique, these guys are often Unskilled, but Strong.
Sometimes, the Extranormal Institute where the hero gets his training has expectations of its graduates that would be too restrictive to make a good story — for example, the final step of the training may be to agree never to use his new power for his own ends. In these cases, the hero being half-trained means he has most of the skill of a fully trained practitioner, but none of the responsibilities — the Cowboy Cop foil to his colleagues' By The Book Cops.
On other occasions, the failure to finish training brings about a defeat. This is not always a Hero Trope: it's often part of the backstory for The Rival or the fate of the Deceptive Disciple. However, it is most often used to create suspense as heroic characters go off to fight battles for which they're ostensibly unprepared. Sometimes the missing training becomes a kind of inversion of Chekhov's Skill, in which the fact that a particular skill is not available to a character comes back at a key moment in the plot. This is part of the hero's journey arc.
This trope frequently turns up in martial arts movies, in the Sword and Sorcery genre, and occasionally in science fiction narratives. Related to The Call Has Bad Reception, which is when a character misses out on critical information or powers because their training was garbled or entirely nonexistent.
See also A Pupil of Mine Until He Turned to Evil, when the training is interrupted by a Face-Heel Turn, and The Paragon Always Rebels.
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Anime and Manga
In the Pokémon anime, Lt. Surge's Raichu had access to powerful Electric-type moves that Ash's Pikachu lacked, but because Lt. Surge had evolved it prematurely, it never learned speed-based moves like Quick Attack and Agility, which gave Ash's Pikachu a decisive edge in their rematch.
In the original Magical Girl Lyrical Nanoha series, Fate Testarossa's combat mage training is revealed to have been rushed and, as a result, very fragmentary, which is why the absolute novice Nanoha is able to catch up with her so fast. Fate's training is only properly rounded up between the first and second seasons.
Kenshin from Rurouni Kenshin left his training because he wanted to fight on the Meiji Revolution, and despite his incomplete training he was still a great warrior and even became the Legendary Hitokiri Battousai; however, fast forward to the Shishio arc, and he needs to complete his training if he wants a chance to beat Shishio.
Used as a Deus ex Machina with Ichigo in Bleach. Also, technically, he never completed his training with the Visoreds.
Surprisingly, this is also the case with Kenpachi Zaraki. His training in swordsmanship was cut short intentionally, as it was feared that if he'd ever been fully trained and decided to rebel, then he'd be unstoppable.
This is a large part of why Gohan struggles in some of the later arcs of Dragon Ball Z; Chi-Chi's overprotectiveness and Goku's absence, he's never fully trained himself in the use of his natural abilities.
Goku himself, on the other hand, is able to get by despite leaving in the middle of his training more than once because he manages to learn everything on his own anyway.
This is also suggested to be one reason that the Supreme Kai is not as competent a King of All Cosmos as one would hope. He was actually a junior Kai back before Buu slaughtered all his superiors, so all the authority fell to him before he was fully ready for it.
Jagi in Fist of the North Star is a villainous example, since only one person at a time gets the complete training anyway. He compensates for his limited skills compared to Kenshiro by cheating.
Psycho-Pass: On Akane's first day on the job, the more experienced Inspector tells her that he can't treat her like a newcomer (since they are understaffed), so she must learn a lot on the job.
The Dragon slayers from Fairy Tail. Trained by dragons that mysteriously disappeared one day while they were children. While they know the basics of their magic, they're mostly Taught by Experience.
The Teen Titans also have this as a premise, fighting things like the demon lord Trigon or the assassin Deathstroke. The Titans are also somewhat notorious for having an incredibly high death rate, which has some Fridge Horror regarding the attitude of the elder superhero community.
The most recent Star Trek film reboots the series with a plot that involves the original series characters piloting the Enterprise against a powerful foe that has decimated Starfleet while they are all still cadets.
Used in Space Camp as the title's campers have to carry out a real space mission.
Luke abandons his training with Yoda in The Empire Strikes Back, precipitating the cliffhanger ending of the film when he enters a fight he's not remotely ready to win. Interestingly, Luke never really does finish his training in the films, and thus doesn't possess the full powers of a Jedi master outside of the Expanded Universe.
Elsewhere, Anakin Skywalker's Start of Darkness is partly related to this trope, since he never learned the emotional control that's part of Jedi training and was thus unusually vulnerable to the temptations of The Dark Side.
In the first film of The Matrix trilogy, Neo hasn't quite learned all the skills Morpheus was trying to teach him in most of his battles. Luckily, the Epiphany Comeback variation of this trope kicks in at the end before Agent Smith kills him. Or technically... after Smith kills him.
Also in that series, a boy who idolizes Neo manages to save everyone using a suit of Powered Armor despite never completing the training program. Captain Mifune, the Zion defense force officer in charge of the entire corps of Powered Armor also admits in his last breath to the same kid that he never finished his training either.
This is how the Bride wins her battles with Elle Driver and Bill himself in Kill Bill, defeating both of them with techniques she learned from Pai Mei which they didn't having failed to complete their training with him successfully.
Morgan and Duncan from the Deryni novels have the ability to heal. Unfortunately there are no trained Healers in Gwynedd to teach them, and the Camberian Council refuses to allow them full training anyway since they're "half-breeds".
The protagonist of The Legends of Ethshar novel With a Single Spell, whose teacher died after teaching him the single spell of the title.
Arguably used in the Harry Potter series, as Harry and his friends are perpetually clashing with much more experienced wizards and witches despite not having finished Wizarding School.
Rincewind from Discworld had to drop out of the Unseen University because one of the eight spells of creation lodged itself in his head and made it impossible to learn any others. An odd example, since he isn't that powerful and doesn't have any real goals beyond finding a nice warm barn to stay in for the night; he's just really good at running away.
While he didn't quite completely drop out, Harry Dresden of The Dresden Files had his mentorship cut short by his mentor turning out to be evil. This has left him pretty much Unskilled, but Strong (especially compared to other wizards) and not too popular with some of the other wizards (after resorting to Black Magic to protect himself from said mentor).
The Star Wars Expanded Universe at times has Luke Skywalker desperately trying to get past this trope, searching for other Jedi or at least surviving documents about them. He's a natural at learning The Force, but that doesn't mean he knows how to pass on what he has learned. He does not find other trained Jedi for decades, but he does find a number of people who got around the same sketchy level of education that he did, and usually ends up filling in those gaps himself. Luke has to find a new way to train others in the ways of the Force, which is why Obi-Wan at one point said that Luke isn't the last of the old Jedi, but the first of the new.
In the novelization of Revenge of the Sith, Yoda and Obi-Wan intentionally decided not to train Luke and Leia in the traditional ways of the Force. The old Jedi failed miserably against the Sith, which Yoda saw as proof that their ways were flawed. Yoda believed it was best to let the Force guide Luke and Leia down their own paths.
Lone Wolf was only a half-trained initiate when the Kai Monastery was destroyed. Unlike some of the other examples, not being fully-trained was a bad thing for Lone Wolf since the powers of a Kai Master are necessary to defeat the Darklords. A good chunk of the first two series involve him finding the material necessary to complete his training.
Live Action TV
In Legend of the Seeker, Richard inherits wizard powers from Zed but must train to be able to use them. However, training takes too long for various reasons and his world is about to be destroyed. He quits, giving up his wizard powers.
Buffy the Vampire Slayer: The Watchers' Council repeatedly tries to claim that Buffy hasn't finished her training, often using it as an excuse to deny her resources or cut her off entirely. As it turns out, the trope is subverted when Buffy realizes her on-the-job training means that she doesn't need them anywhere nearly as much as they need her as an active agent.
It's played straight to some extent with some of the potential slayers in Season 7, however.
During a few early missions of the Weapon Master Mode of Soulcalibur II, you're attacked by bandits. Your mentor sighs and notes that few complete their training under him, going on to become bandits in the region surrounding his school during one.
Dan Hibiki from the Street Fighter universe is a Joke Character variant of this: His master told him that he had nothing further to teach him after learning that his main motivation was to avenge his father, which Dan took to mean that his training was complete. He gets his ass kicked constantly by the others as a result of this.
The above example from the Pokémon anime also applies to the games as well. While evolved Pokémon learn moves at later levels than their base forms, stone-evolved Pokémon such as Raichu typically stop learning moves at all, apart from those taught by TMs/HMs and tutors. Prematurely using an evolution stone on such a Pokémon can therefore permanently lock that Pokémon out of learning their most powerful moves.
In Kingdom Hearts I, the 14-old Sora (whose combat experience consists of sparring matches on the beach) is given a magic key-sword and told to save the universe. Fast-forward a few games and we learn that Keyblade wielders used to be The Chosen Many, with years of training before they even started world-hopping. Thanks to the Big Bad, all of them were incapacitated except King Mickey, who prefers to work from behind the scenes. The plot of 3D is kicked off by Yen Sid's decision to formally retrain Sora and Riku.
In Mass Effect, Kaidan Alenko and Jack both got their biotictraining shut down before it could be completed. When Kaidan enlisted, the Alliance offered to pick up where BAat left off, but Kaidan had enough residual trauma that he turned them down in favor of training as a technician and a medic. Jack didn't have anyone who could have helped finish training her biotics, so she taught herself how to use weapons instead. Thus, despite being extremely powerful by human standards, neither is an Adept.
Shirou Emiya, as Fate/stay night notes in its opening, is an Ordinary High-School Student who was taught a smattering of Magecraft by his adopted father before he died. In the Nasuverse, where the indicator of a mage's power is determined by how many generations a bloodline has been dedicated to maniacally researching and enhancing their skills, "incompletely" trained doesn't even begin to describe a boy who's been practicing some half-assed pointers in his spare time. Depending on the route of the story, and the epilogue, Shirou becomes a legendarily powerful warrior, either way.
Fighter from 8-Bit Theater suffers from this... with the additional problem of hardly ever having listened to his teacher.
The Ben 10 series. Ben's training has pretty much been what his grandfather shows him, and figuring it out on the fly, even though other kids with Plumber-related abilities get to train properly at the Plumber Academy. This was eventually rectified in Ben 10: Ultimate Alien as Ben, Gwen, and Kevin ended up receiving proper Plumber training and passing the courses.
Kim Possible had Ron Stoppable not completed his training due to lack of self confidence and a phobia (the style was Monkey Kung fu, and Ron has a fear of monkeys). This kept him at the Bumbling Sidekick level until the Series Finale, at which point Ron got it and stepped up.
Parodied and double-subverted in an episode of The Simpsons when Bart leads his peers to Shelbyville to recover a stolen lemon tree. A subject he ignores during school early in the episode — Roman Numerals — becomes critical to freeing him from the tiger cage at the Shelbyville zoo. Despite complaining, "They never even tried to teach us this in school!", he ends up using his knowledge of the Rocky sequels' titles to deduce the answer anyway.
It's played straight in another episode of The Simpsons when a newly-sober Barney, still training on a helicopter, is forced to fly with Homer to save Bart and Lisa from a forest fire.
In Avatar: The Last Airbender : This applies to most of the cast at various points. Aang may be a master airbender, but he struggles to master the other three elements, and even at the end of the series it's noted he still could use some work in Earth- and Firebending. Katara is a talented Waterbender, but until she finds a master to teach her in the first season finale, she's only capable of basic bending. Zuko spends much of the first season still mastering the basics of Firebending. And Sokka has no bending, and while he's skilled in using his club and boomerang, he says he feels useless compared to his bending friends. The only real exception is Toph, who joins the group already a master Earthbender, and even invents a new form of bending down the line.
Admiral Zhao also never bothered to complete his Firebending training, as he skipped the discipline part. Aang, upon learning this from his former mentor, promptly uses this knowledge to bite Zhao in the ass during a fight.