Why do some people hide how smart they are?

1. A lot of very smart people are convinced that they aren't! It's called impostor syndrome. [1] They aren't trying to hide it so much as doubting it's actually there. As a result, they try to downplay it so as not to raise expectations.
2. Sometimes they find it advantageous. This is particularly true for women because, sadly, some men are intimidated by intelligent women.
1. It's generally counterproductive to have coworkers/managers/etc intimidated by you, so some very bright women pretend to be a little less bright to avoid negative consequences. That's right, they fear negative consequences just for being smart. It's a problem for men too, but far less frequently.
2. I've had very smart women complain to me that dating is harder for them. They aren't just being dramatic. I've seen it happen. Men are taught that they should be the strong one’s in the relationship and some men can't handle being in any way inferior to their partner. To every bright woman reading this: avoid these men like the plague! Unfortunately, many intelligent women feel they have to play dumb to garner more favorable male attention. I actually had a girl tell me in high school that she although she wanted to do medical research, didn't want to major in biology because “guys don't like smart girls”.
3. They are taught to from a very young age. Kids make fun of other kids for being different, so smart kids start to hide the fact that they are different. They stop asking and answering questions in class. They don't enter their hovercraft in the science fair. It becomes habit out of necessity.

This question reminds me of Douglas Adams’ books, where the ultimate answer to life, the universe, and everything was 42. When the people insisted there must have been a mistake, the computer suggested that perhaps they didn’t understand the question. (And then they made an even more powerful computer to find the ultimate question that the ultimate answer answers.)

So while I see issues in the wording of the question, I might, however, understand the answer you’re looking for.

Say you have a set of skills that can be useful. If you’re boastful and shove those skills in everyone’s face, people see you as arrogant, a braggart, a jerk, etc. Even if you don’t, some people can feel threatened because they don’t have those same skills and don’t feel they can compete. Sometimes there are outdated social values still being perpetuated. (Like where a man feels like he has to be in control, otherwise he’s not a man, never mind that equal partners means not being dominated as well.) Can you see why someone would try to mitigate the tone of their behavior, so they don’t come across in a way they don’t wish to?

As for the wording, I tried to deconstruct the word “smart” or “intelligent.” What I came up with is either “I like the way you think” or “I believe you think the way I believe people should think.” When you look at it like that, the word becomes meaningless, completely subjective, almost emotive. Take the television caricature of “intelligent,” Sheldon Cooper. In the show, he has several skills. Social skills are not among those. If intelligence was an objective thing (and if he was real), I would hope he would have the sense to realize and inform people that he has Asperger’s syndrome, and to compensate by adopting values to base actions on that create synergy with other people in spite of the disorder. But, this is “intelligent” in the eyes of the show’s writers. (I seriously dislike how his character is written. He does nothing to soften his image, and he comes across abrasively.)

While it’s true that some people actually hide their smarts, for a variety of reasons (many addressed here by others), might I point out that many simply don’t sit around tooting their own horn. For one thing, smart people understand that they’re smart in some areas and less smart in other areas and downright dumb in still others. Intelligence is far more variable and complicated than we’re often led to believe. What’s really smart is understanding that joining with others to create or problem-solve is far more effective and rewarding than working alone. It generally doesn’t help working relationships to view yourself as the smartest person in the room—even if you are. It’s far better to put people at ease so they feel comfortable about bringing their own expertise to a project. Much better to focus upon intelligent cooperation than to compete for the title of Who’s Smartest.

That said, I definitely understand the impulse to “dumb down” if you’re a woman (many women do so to reduce the abuse heaped upon them). But I began to notice that the few girls who refused to pretend, while they might not have dated as much as many others, ended up dating the guys who liked smart girls. They ended up dating the confident guys. They seemed to spend far less time being aggravated with their romantic relationships.

Now in my 7th decade (good grief!), I enjoy the fact that there’s no need to pretend with my spouse. He knows the truth of who I am and accepts all the strangeness and dumbness along with the bright. He’s confident, and accepts his own frailties while embracing his own intelligence. Because we know each other and accept whatever’s true about ourselves and each other, we have little difficulty cooperating with each other to accomplish what we choose. And it’s so, so nourishing to simply live honestly with someone, spending zero time pretending to be something you’re not. Pretending to be something you’re not is unlikely to attract the partner who’d best suit you.

Speaking from personal experience: I knew I was of greater than average intelligence, but rejected the “genius” appellation four a couple reasons:

1. It sets unrealistic expectations. High-IQ individuals (I’m around 150) do not generally live up to the societal stereotype of the “genius.”
   
   One can, for example, have very high marks in linguistic skills, abstract thought, pattern recognition, and memory, but suck at math. I don’t suck at math, and can pound through most arithmetic and basic algebra, but I get tripped up translating word problems to numbers, and while I understand higher-math conceptually, I can’t “do” it very well.
   
   If one caries the “genius” stigma, everyone expects you to be brilliant all the time. This is not the case, and it creates a lot of unnecessary pressure. Gifted individuals still make plenty of stupid mistakes.
2. If you have a high IQ, and also a reasonable level of emotional intelligence, you will recognize that your intelligence can act as a barrier in a number of ways. For example, in personal interactions, you have to learn to perceive what people mean, regardless of what they actually say, and learn not to react as your inner world compels you to.
   
   On one level, some high IQ people really are clueless. Even when they know full well what the other person meant, they are impelled to correct them, and are unaware how arrogant they are perceived to be. On the other hand, some of them are just jerks. They know they are arrogant, and simply don’t care. Such people, despite their potential, tend to poison their relationships.
   
   But for those that do care, and are self-reflective enough to recognize the way others perceive them, they will learn to hide their intelligence for the sake of peace, sanity, and friendship.