Kids have lots of great qualities. Energy, creativity, candor, and - overdone as it sometimes is in fiction - innocence and openness. Like all people, they vary, so even these traits don't apply to all children, but one thing that just plain doesn't belong with children? Wisdom.
Wisdom - the kind that can make a person empathetic, patient, a good judge of character, and knowledgeable about life, truth, and relationships - comes with experience. Once again, this isn't cut-and-dried; an older person may not be more patient or empathetic than a younger one. But think about it: it takes a couple of years before a child is even capable of empathizing, of realizing that other people even have minds, internal lives, wants, needs. In general, children and teens have shorter attention spans, are less thoughtful, and have shakier, more selfish, or more myopic judgment than older people of similar backgrounds.
None of this means that I think people aren't writing child characters impatient and selfish enough. (Although recognition of these flaws can be a powerful tool for getting reader sympathy. I love that Diana Wynne Jones' characters often have selfish wants, even if they are too ashamed to express them.)
No. What it means is that (human) child characters need not to smile knowingly and state simplified universal truths, leaving the other characters in awe. They need not to instantly recognize relationships between other characters, and especially not to smirk and tell the protagonist how obvious it is that s/he likes the romantic interest, well before anyone else catches on. They need not to immediately recognize dishonesty or untrustworthy persons. I don't care if they are streetsmart young members of the thieves' guild or classically educated princesses in courts full of intrigue or blind kids who are super-duper practiced with using their other senses. They must not be Wise in the Ways of the World.
Ideally, you also want to be careful with younger child characters who open their eyes really wide and say, "Mommy, why do people hurt each other?"
Kids can certainly be know-it-alls. Hermione is a great example of this, and one of my favorite fictional brainiacs. Her know-it-all-ism is much like mine was in school: a lot of book-smarts plus a desire to prove herself plus some social awkwardness. She is, in that way, quite a realistic character, as well as sympathetic and entertaining.
Of course, kids and teens may think that they are wise when they aren't - as may anyone. And like anyone, they can have moments of insight or brilliant realizations. Just don't make them frequent, and don't make them enough of a distinguishing characteristic of a child character that one could call the character "wise."
In part, this just comes down to the fact that it's okay for a child character to have more knowledge and/or technical skill than I do, just like it's okay if a child character can use magic when I can't. When a child character has more wisdom than I do, it sets off both my BS-detector and my growly face. I resent being told how life works by people who have experienced very little of it.
I'm sure that, like any rule, this has exceptions, and I welcome hearing about them!
On an unrelated note, I'm not sure whether any of you are doing NaNo this year - if so, good luck, and do tell! - but here is a most entertaining little explanation and endorsement of the phenomenon.