The truth here is the fact that education doesn't equate to intelligence. One can read all he/she wants but if those things read are not put into action or your everyday routine, there's a high tendency that such a person won't prevail.
I agree on your first statement. But that would depend on how your education system works. Take our country for example: Teachers are underpaid and would not get the benefits that they should be getting. Hence, lowering some of the teacher's morale on educating the kids properly which lowers the kids' quality of education. It's a domino effect.
But then again, intelligence sometimes would be innate to a person. It would somehow become natural depending on how they were raised, I guess.