Am I a man or am I a muppet? If I'm a muppet then I'm a very manly muppet!
Let's get this clear... a "broken" game is one that, played by the rules as written, does NOT WORK as a game. It locks up, it spirals needlessly until the players long to run from the table, it creates situations that keep the game from reaching the conclusion intended.
"Broken" is not "inadequate development", either - while I will never play Vox Populi again (due to woefully bad playtesting/tuning of the action cards), that doesn't mean the game is unplayable - it's just unbelievably irritating to play. (I feel the same way about the end game of Krieg und Frieden.)
So, with those two things out of the way, the question remains: what is a "fragile" game?
Since it's my blog, we'll go with my personal definition:
A fragile game is one that is highly susceptible to newbie and/or substandard play.
And with that description, I realize that my personal frustration with fragile games is in inverse proportion to their length and in direct proportion to their gamerly qualities.