So Near, Yet So Farfetch'd - S1-E47
Continuity mistake: When Team Rocket takes off in the boat, it sinks. Their pokeballs float away, and Jessie says "Arbok and Lickitung are floating away!" But she doesn't catch Lickitung until Princess Versus Princess (Season 1, Episode 52).
Suggested correction: It's because these episodes were aired out of order.
Corrected entry: Ash suddenly has a Charmander, but his original Charmander evolved all the way to Charizard a long time ago, and we never see him get another Charmander. He couldn't have gotten another one, as we see each and every single pokemon that he catches within the show. Even in the Japanese-only episodes, he catches a bunch of Tauros, which Oak mentions when he returns to the lab.
Correction: That was a banned episode, which was before Charmander evolved. Http://bulbapedia.bulbagarden.net/wiki/Snow_Way_Out.
Wherefore Art Thou, Pokémon? - S2-E19
Corrected entry: In the recap it says that Ash won a "big victory over Pokemon Master Prima", but it was actually a crushing defeat. (00:01:08)
Correction: It was presumably done as a joke.
Mystery at the Lighthouse - S1-E13
Other mistake: When Brock turns his head to the costume, his nose is missing. (00:12:47)
Suggested correction: Agree with the other correction. Dropping details, such as facial features like a nose or mouth, and simplifying character designs is a very common and encouraged drawing/animation technique to imply distance. This isn't something the animators would just overlook. It was a deliberate decision. Ash and Misty similarly look somewhat less detailed than they do in closer shots. At best, this would be a deliberate mistake, but I personally think it falls under the umbrella of "suspension of disbelief" and shouldn't be considered a mistake given it's a stylistic choice to imply distance.
Suggested correction: His nose isn't seen because he is far away from the camera.
Corrected entry: When Meowth comments that Victreebel's Sleep Powder works well, James closes his eyes, tilts his head up, and says "Yes, we're a knockout combination." As he says this, his nose inexplicably grows to about ten times its normal length, but goes back to normal after he says the line.
Correction: It's pretty obvious that this was done on purpose for comedic effect (growing noses are usually an homage to "Pinocchio"), and thus is not really a mistake. Remember, this is a hand-drawn show, so it's not like the animators would simply overlook a very noticeable detail like this.
Corrected entry: Krabby evolves into Kingler during the match, and is somehow able to use Hyper Beam. Krabby can't learn Hyper Beam, and Kingler can only learn it via a Technical Machine.
Correction: Differences between games and anime were fully intentional and they're not mistakes at all.
Primeape Goes Bananas - S1-E25
Corrected entry: When Ash is talking to Professor Oak, the latter tells him that Gary has already caught 30 pokemon. But in the episode Mystery at the Lighthouse (Season 1, Episode 13), Oak said that Gary had already caught 45 pokemon.
Correction: The episode aired out of order due to controversy caused by an earlier episode. Not a mistake due to the Pokemon being correct in the order it is supposed to be in. On DVD releases, the episode is placed correctly.
Pokémon Contest! Hajitsuge Convention! - S7-E10
Factual error: During the battle with Team Rocket, Beautifly uses Tackle. However, it was unable to learn it until Generation VIII. This episode, a Generation III episode, would mean that at this point in time, Beautifly was unable to learn it by any means. (00:19:57)
Suggested correction: Differences between anime and games are not mistakes.
Factual error: During the second battle with Team Rocket, Victreebel uses Tackle on Ash and his friends. However, it cannot legally learn it in the Pokémon games. (00:18:18)
Suggested correction: Differences between anime and games are not mistakes.
Flower Power - S3-E6
Factual error: During the second battle with Team Rocket, Arbok and Victreebel use Tackle, a move they cannot legally learn in the games. (00:18:15)
Suggested correction: Differences between anime and games are not mistakes.
Chicorita's Jealousy!? - S3-E20
Continuity mistake: At the beginning of the episode, Chikorita is inside a pipe, but in the next shot, the pipe vanishes. (00:01:16)
Suggested correction: I watched the shot and Chikorita is not in the pipe to begin with.
Character mistake: When Mandi brings out Golbat, Brock says, "Kingler is really weak to air attacks." But Kingler's a Water type, which has no weakness to Flying-type attacks.
Suggested correction: Brock is saying that Kingler is vulnerable to attacks from the air, not that it is weak to Flying-type moves.
Continuity mistake: When Ash says, "Misty'll take good care of your Marill, you'll see," Wilhomena is holding her parasol with on hand, but in the next shot, she is holding it with two. (00:09:16)
Suggested correction: I rewatched the shots, and Wilhomena is holding her parasol with both hands in both of them.