Honestly it depends on how big the weak spots are, and how many of them there are. If you build Captain I Suck at Everything in Life But Punching People, then odds are you are gonna possibly get screwed by those weaknesses everytime you are doing something other then punching people.
The more holes in a character, the more flaws they have, the more situations that they will be exposed.
All that being said, honestly, to dog a character for flaws in it constantly is no fun. As a GM if I had so many issues with a character that I was exploiting its flaws every single session, there is no way that I would have allowed it in my game. Not only is exposing the one dimensional character no fun for the player of that character, but while you are working over that character you tend to be ignoring the other players at the table.
Ultimately while a player may ask to play a concept that a GM does not like, the GM has final say over the concept. Why the hell waste time in game punishing a player for a character you don't like, when could could have just bit the bullet and said NO in the first place, or worked with the player to make the character viable in your world.