The United States has made great strides over the past few decades in increasing the educational attainment of its populations. Millennials are the best-educated generation in American history and the baby boomers were themselves much better-educated than the "greatest generation." This means that the share of high school dropouts has greatly declined and the share of college graduates in the population has sharply increased.
One consequence of this is that we have massively improved the educational credentials of people living below the poverty line, as shown by this great chart from Matt Bruenig of the progressive think tank Demos:
During this time, the overall poverty rate has risen by 1.1 percentage points. This ought to cast some doubt on the idea that further increases in educational attainment are going to cure poverty over time.
Bruenig offers some thoughts on why the education cure hasn't worked, but I think he overcomplicates it somewhat.
The key point I would make is that the poverty rate facing people who have full-time jobs is actually pretty low:
People face various kinds of barriers — the macroeconomic situation, economic conditions in the town where they live, certain kinds of disability, family responsibilities, substance abuse problems, etc. — that make it hard for them to get a full-time job.
To reduce poverty, you either need to address those barriers or you need to just hand over some money. More schooling has certain kinds of real benefits to society, but it hasn't moved the needle on poverty historically, and there's no reason to think it will in the future.