I went to a posh school, but that is totally not the reason why I am, arguably, smart.
The headmaster there gave all of the students a speech about how we must learn to think critically. Think for ourselves. Take in the information around us, assess it, think about it rationally, and be honest with ourselves in how we perceive it.
That's it. That is being smart in a nutshell. Smart people do that. Anyone can do that. You just have to understand that the truth isn't always what you would like it to be... and whatever you find, you should question it, and test it.