Something that many gifted children struggle with is perfectionism, especially in relation to academics. See for example Perfectionism and the Gifted Child at Hoagies' Gifted Education Page. There are a number of possible reasons for this, but one can be that their sense of identity becomes heavily dependent intertwined very well (achieving near perfect performance), and this can lead them to feel that if they don't do very well, then they're worthless.

Sometimes the standard that they judge themselves against is a relative one: e.g. being ranked #1 in a high school graduating class. Sometimes the standard that they judge themselves against is an absolute one: e.g. getting a perfect score on a math test.

It seems to me that it helps to realize that in the long run, one's performance will almost certainly fall far short of perfection, regardless of who one is.

The academic environment that a child is exposed to is usually artificial in a sense. For example, exams are designed so that all of the questions are doable with a reasonable amount of study, such that it can be possible for gifted children to get perfect sores. One could construct much harder exams, such that even a highly gifted child would score only 50% or lower.

Even if being the best student in a K-12 class is within the realm of the possible, one will almost inevitably end up in a context in which one isn't exceptional. It's common for valedictorians to go on to Ivy League schools and to be shocked to find that they're only average within that population. There are 4 million American high school students of a given age. Even if one is the best student in a class of 400, the a priori probability of being the best student in the country is 1 in 10,000.

Even if one is the best student of a given age in the country (for example, in mathematical research ability), if one is ambitious, one will still fail routinely. The Riemann Hypothesis in mathematics has been unsolved since it was first 1859. Five+ generations of mathematicians have grappled with it without success. Even if one is the among the best mathematicians in the world, and works on it full time, one probably has a < 10% chance of solving it.

Being exposed to this perspective can initially be jarring. As a matter of reality, one almost certainly isn't going to be the best in the world at something hard, and one almost certainly isn't going to do something regarded as amazing. Almost all of those very smart people who dream of proving the Riemann Hypothesis will be disappointed. But once one accepts this, one can learn that it's possible to have a happy life anyway. The sooner one comes around to this view, the better. It can be very liberating.

For those of you whose children are perfectionists, does this perspective seem like it would be helpful?
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Advising for gifted children available at Cognito Mentoring.