This is one of those statements that really bother me, almost as much as someone telling me I need to prove the non-existence of God if I want to be an atheist. Yes, intelligent people should have a good understanding of how big and complex the universe is and how little they know compared to the total amount of knowledge available. And given that they are aware of such, it wouldn't make sense for them to walk around claiming they are a big deal. However, people do not always act rationally, even geniuses. There is no negative-correlation between claiming to be smart and being smart, in spite of the obvious contradiction. Claiming to be smart and being cocky about it, it's just a personality trait. Think about this: you IQ is defined and practically unchangeable by the time you are a young teenager, but your personality continues to develop for many more years after that. So there is no way your personality can affect whether you are smart or not.
Just a thought though.