Continuity mistake: When Ash & co. are in the boat with Team Rocket (who's dressed as Vikings), watch Brock's shirt sleeve (The one away from Misty) when Misty puts Togepi inside her knapsack. It goes from a dark orange to green (the same color as Brock's vest) for a few frames before turning back.
Other mistake: When the Poké Balls containing the Trainers' various Pokémon are sent to the DNA Cloning machine, Team Rocket incorrectly identifies Scyther as Alakazam. This is a dubbing error as, in the original Japanese version, the Rocket-dan (as they are called) correctly identify the Pokémon as a Strike (Scyther's Japanese name).
Other mistake: In Mewtwo Strikes Back, when the pirate trainer sends out Machamp (the four-armed Pokémon), the animation of it appearing is shown twice. Also, the sound effect that plays when a Pokémon appears isn't played the second time. There was a small section cut out of Ash throwing Squirtle's Poké Ball in the air. That was replaced with Machamp coming out twice.
Character mistake: As Team Rocket discovers the cloning machine made by Mewtwo, they start naming the shadows of each Pokemon as they appear on the screen. As the first Pokemon appears on screen, they say "Alakazam" but it is actually an outline of "Scyther" who is a different type of Pokemon.