Just to be sure I looked up the official definition of wisdom and, sure enough, it says,
"the quality of having experience, knowledge, and good judgment"
So I guess that part of the answer to the question is yes, you become wiser as you grow older. But there's another aspect to the idea of being wise and that's paying attention to the experiences you have, using the knowledge you've acquired, and exercising good judgment. Adults seem to have an 'override' key that they constantly use to bypass what experience has taught them, ignore the knowledge they've acquired, and leave good judgment out of an equation. While at first glance that appears to be a foolish thing to do, at second glance it's why humans are creative, innovative, and industrious. Think of all the advancements we have in technology, medicine, aeronautics, and every other of human endeavor we wouldn't have if everyone always exercised wisdom.
My answer to the second part of the question is yes, we can always learn something from children--no matter how old they are or how old we are. They see the world through eyes that we are no longer privy to so we miss some very important nuances.