THE FLY
If you're wondering why you're getting that weird feeling that you've seen Bill's story before, it's because you most likely have. Bill's accident is actually one of many homages to a single movie, the 1986 remake of The Fly. To clarify, there was an older film in 1958 that was meant to be the original, but really, they're practically similar in name only. While it could be possible that Bill's story (among other homages) could have been based off the earlier version, it's easier to assume that it's a reference to the 1986 film because the storyline wasn't prevalent in pop culture until after its release.
In brief, the film centers around the brilliant but meek Seth Brundle, a physicist who was on the verge of perfecting a matter teleporter. However, he becomes involved in a relationship with the journalist Veronica Quaife, to whom he confides to avoid going crazy with trying to keep his invention under wraps. As the two grow more intimate with each other, Veronica's past (in the form of her ex-boyfriend and current superior, Stathis Borans, practically stalking her) resurfaces, which she attempts to hide from Seth. The latter grows paranoid that Veronica is cheating on him, and in a drunken state, he decides to teleport himself while she's absent from his laboratory. Unfortunately, during the process, he's unaware that a fly has slipped into the teleporter pod with him, and as a result, the two are transferred and merged. The film takes a gory turn from there as Seth gruesomely transforms from his normal state to a half-fly, half-human monster over several months. He searches desperately for a cure, and Veronica watches in horror, right up until his final transformation. Sanity shaken and body irrevocably transformed, Seth ends the film on a not-so-happy note.
Despite its dark tone and gory effects, The Fly was seen as a sci-fi icon. It won multiple awards ranging from various Saturn Awards to an Academy Award for its special effects. At the same time, it had a considerable fanbase in the 80's, and its unique storyline has been mimicked by popular culture from Teenage Mutant Ninja Turtles to Invader Zim.
Hence, Bill's story is actually a reflection of Seth's. However, because Pokémon isn't nearly as dark as The Fly, several noteworthy changes had to be made to the story to make it a little less horrifying for the games' main audience.
First and foremost, The Fly was actually a metaphor for debilitating disease. Seth's transformation was slow, painful, and horrifying, taking place over several months. Although he tried to find a way to reverse the process, ultimately, he could only watch as he gradually lost his humanity. Conversely, Bill's transformation tended to be off-screen and instantaneous. We're never shown what he looks like during the process, and we're led to believe that he didn't feel pain from it. This change takes the edge off what would otherwise be a rather gruesome ordeal, changing the main point of the story from a heartwrenching and shocking analogy to simple, light-hearted sci-fi.
Second, Bill's accident was less out of karma as it was out of his own clumsiness. In Special, he unintentionally started the teleporter, whereas Seth's reasons for teleporting and not noticing the fly were rooted completely in his fairly human urges. To be a bit more specific, Seth teleported due to his drunken state combined with his jealousy and paranoia towards his lover. This fact adds more of a human element to the accident in that Seth was acting out of his own impulse. Meanwhile, the accident in the Sea Cottage was completely an accident, meaning it becomes less serious simply because Bill doesn't always notice all the little details he should be focusing on.
Third and most importantly, the story obviously comes to a happy ending. Whereas Seth, in his Brundlefly form, loses his sanity and begs for Veronica to end his life, Bill is able to maintain his normal personality even in Pokémon form. Because of this, he can create a program that separates himself from his Pokémon, and he can enlist the help of a passing trainer to activate it for him. In other words, while The Fly describes a descent of man into horrifying disease, Bill's story, due to the nature of Pokémon, ends on a lighthearted note with Bill being able to turn back into a human and walk away from the incident as if nothing had happened.
With all three changes in mind, Bill's story falls into the same category as a lot of other homages to The Fly. Although they share the same basic plotline (scientist merges with a non-human via accidental teleporting to become a monster), it's ultimately a far more lighthearted sequence with a focus shifted towards the sci-fi aspect of the film (as opposed to the metaphorical parts) simply because the original is intensely dark.