Correlation != Causation
What does this mean? Essentially it means that because two things are corellated to each other does not mean that one begets the other. The notions of "Good Schools" vs. "Bad Schools" seems to imply that the school itself is the major contributing factor and influence to a childs education. Many studies, politicians, and everyday people espouse this but it may not be reflective of reality.
Poverty affects childrens brains
According to this article about this study done in 2008, they have found that the brains of impoverished children 9-10 compared to wealthy children of the same age have such stark contrasts that it is as if comparing a healthy brain to that of a stroke victim. Environment and diet play an enormous factor when it comes to a child's brain development.
Another contributing factor that may play into a child's performance at school is parental involvement.
Does parental involvement influence the academic achievement of Mexican-American eighth graders? Sorry paywall, but the abstract summarizes the findings
This study finds a significant corellation between parental involvement and better educational outcomes for their children. It stands to reason here also that middle and upper class parents have more time and ability to be involved in their children's education than say an impoverished family working multiple jobs.
So the final correlation to tie these two together is in a study that finds that socioeconomic index is the most likely indicator of negative school outcomes.
So I would reason that such a hypothetical claim in light of this evidence doesn't stand to reason. The school itself has little influence on your education, it has to do with your family and community background that you grew up in.