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"Sometimes I wish Doctor Robotnik hadn't made us so persistent."
"He made me persistent; he made you too dumb to quit."
—Grounder and Scratch, Adventures of Sonic the Hedgehog

Sometimes there's a character who just does everything wrong, and then some. When they're being chased by a cold-blooded murderer, they're more likely to run into his arms and whimper for mercy than, let's say, run away. When a flood/avalanche/rock slide/truck comes towards them, they're more likely to stand there gawking than run. When a villain has kidnapped them and has them dangling over a pit of acid, they're likely to taunt the villain and piss him off (and probably reveal that the hero is coming as well) to the point where he decides to kill them anyway, rather than staying quiet, content with the knowledge the hero is coming.

This character is Too Dumb To Live.

Defying the laws of logic, they always seem to survive anyway. The hero always busts in at the last minute to save the idiot, saving the day and earning the dumb one's eternal gratitude.

And then the dumb one goes out and does it again.

And again.

And again.

Common in continua such as Buffy the Vampire Slayer, Smallville, Invader Zim, and Star Wars (basically the whole Jedi order in that instance. Don't be fooled: any group whose members can sense darkness but can't sense Senator Palpatine's until it's too late, not to mention Anakin Skywalker's Dark Side shift, doesn't deserve to be revered; and let's not forget they have a series of utterly ludicrous laws that directly go against human nature and stop them from building up a group that can defend itself numbers-wise).

Also a common trait among Mary Sues, but, just like on television, the intelligent canons always forgive them.

See also the page with the same name on TVTropes, which describes pretty much the same thing.