Generally, after a lifetime of honest, hard work, you'll make it 1, maybe 2 tax brackets higher than your parents. To make it further than that you have to be especially talented, lucky, or both.
I just looked it up: as of 2016, 7.5% of kids born in the bottom quintile make it to the top quintile, so it is indeed possible, though quite rare. [1]
I think your basic sentiment is right, though you'd probably need to insert one more generation: a poor kid has everything they need to make it to the middle class and live a not-poor life, a middle class kid has everything they need to make it to the top 10%, and a top 10% kid has everything they need to become fabulously wealthy.
The poor -> middle class leap is probably more difficult than any of the others.
[1] https://www.economist.com/graphic-detail/2018/02/14/american...