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Dragon Ball Z
A general one for Dragon Ball Z. Satan/Hercule somehow convinces everyone that not only was Cell simply using bombs when he destroys the military on camera but also that all of the real fighters at the Cell Games were simply using tricks. There are two problems with this. The first is that we and the people watching the video clearly saw Cell being hit by missiles multiple times to no effect. Bombs do not grant near-invulnerability. The second is that this is the same world where, only a little over a decade ago, Goku, Kuririn, Tenshinhan and many others were filmed live in a major tournament using those same 'tricks' with the announcer and the audience clearly aware of what those 'tricks' were. This is the same world where Master Roshi (as Jackie Chun) blew up the freaking moon and King Piccolo took over the planet. Even if you want to argue that those events all took place in the early manga which was far more lighthearted it still doesn't make any sense because Nappa clearly blew up a camera crew by pointing at it and at the time of Satan's claims Cell has just finished consuming a vast portion of the human population. Has everyone taken collective leave of their senses?
Vegeta and Gohan's respective lapses of logic during the Cell saga certainly make one want to smack oneself given how after the events of the Freeza saga both should know better. Both of them should know first hand not to underestimate Cell: Vegeta knew just how dangerous a powerful enemy could be and that it was best to finish them as soon as possible, while Gohan knew a foe could pull something really nasty if desperate enough given how Freeza reacted when he was outmatched while fighting Goku. But Vegeta instead helps him get stronger and Gohan later proceeds to spare Cell because he's having too much fun dragging their fight out regardless of the consequences. Krillin's move of sparing #18 can be viewed as this, but it can also be somewhat justified since he may have assumed that Vegeta wasn't holding the Idiot Ball at the time and would destroy Cell before he absorbed her rather than helping him succeed in the task instead.
During Goku's final battle with Piccolo at the end of the original Dragon Ball series, Kami tries to help him out and Goku (in the Funimation dub, at least) responds by saying "This isn't your fight, Kami," and then giving Piccolo a free shot at him to make things "fair". Given that Goku was literally fighting Kami's evil half, one would think that this was a hell of a lot more Kami's fight than it was Goku's. Goku was completely out of line, yet the battle ends with Kami not only admitting to his righteousness, but offering Goku his job!
Goku giving Cell, an active threat to the world at large, a Senzu bean before sending Gohan to fight him. The world IS at stake here. Fuck fair fights!
Uub's placement in Dragon Ball GT. After Goku left for ten years to train him, Uub returns in the Baby arc. So what does he do? He gets beaten by Baby, fuses with Buu to become Majin Uub, gets beaten again by Baby and eaten, paralyzes Baby long enough for Goku to kill him, gets beaten by Super Android 17, and finally does nothing while fighting Omega Shenron. Ten years of training, huh?
And that's not even the worst part. The worst part is how he's the third strongest "hero" character (After the two Super Saiyan 4), yet everyone treats him like if he was Yamcha.
Here's one in the Buu Saga...Gotenks and Vegeto are enjoyable, but one can't help but wonder why these two would fool around when battling Buu when the Earth was at stake and Super Buu killed/absorbed their friends and family!
Unfortunately, Fusion seems to bring out the worst traits of the combined characters it seems like. Granted Vegito wanted Buu to absorb him so he could free the others but Gotenks could've defeated Majin Buu when he was Fat Buu but he would rather show off and make an idiot out of himself that he got creamed. Bad enough in the rematch against Super Buu instead of going into Super Saiyan 3 immediately he decides to waste time and perform numerous moves that have zero effect on him and it takes Piccolo nearly about to strangle him to get him to fight seriously but by then his fusion time is up. If he had done so, Buu would've been destroyed and not escape the Time Chamber and eat the rest of the Z Fighters.