Basic Trope: A character who is incredibly smart, irritatingly aware of it, and driven to go beyond himself, which often leads to spectacular failures.
Straight: Princess Velvet is a bright, canny young woman with a knack for diplomacy and a magical prodigy. Her greatest weakness by far is her Pride, as she tends to overestimate just what she's capable of...
Downplayed: Velvet's very proud of her intelligence, but isn't particularly boastful about it; mostly she just has a way of smiling as she lays out her plans and watches them succeed. When one of them doesn't work the way she intended, and she has to watch others scramble to recover on the fly, she becomes very, very quiet.
Justified: Velvet has to excel just to be allowed onto the same playing field as her (mostly male) peers; this also feeds her arrogance as she considers herself superior to the more misogynistic members of the lot, and her drive to push the limits and prove her superiority.
Velvet appeared to make a mistake, but she made a mistake intentionally to trap Evulz.
Velvet really is just as good as she claims to be, and displays intelligence equal to what she believe she had.
Double Subverted: However, Evulz managed to turn her fake mistake into a real one.
Zig Zagged: Sure, Velvet's smart — smart enough to know her limits. But then Emperor Evulz makes it personal and provokes her into making a serious mistake... that turns out to have been a feint on her part, making it look like she was making a grave error in judgement. However, her success at turning the tables gives her quite the reputation, and while she claims not to care, she does start growing increasingly smugger and more self-assured...
Averted: All characters know the limits to the smarts.
King Sable drives his daughter to excel simply to earn the right to continue her studies, as he fears that she'll make herself a target, yet is too wishy-washy to put his foot down so long as it's clear she really does benefit from it.
Queen Satine encourages her daughter's superiority complex because she enjoys having an unparalleled genius as her child.
Defied: King Sable and Queen Satine teach Velvet not to lord her intellect over others, as it's not good for a future ruler to alienate others.
Deconstructed: Velvet's arrogance further isolates her from her peers, including those who would gladly befriend her given the chance, and makes her a target. After she fails spectacularly, her enemies use it to completely ruin her reputation. Velvet is also shaken by her failure, and never manages to escape the specter ofher biggest mistake.