No, 3-4 generations is about right. The first generation needs to work hard just to ensure there is enough food for the next. Without enough food everything else is done: intelligence is greatly harmed by not enough food. That Food then lets the next generation work a little smarter, and send their kids to school instead of the kids having to work as well just to ensure there is enough. The kids now have some education to work with, but unless they get scholarships it isn't enough to get ahead, but it is enough to save for their kids scholarships. There you go, 3-4 generations to get out of the hole - assuming everyone all works hard to get out, anyone can drop back (either from laziness or bad luck).
Luck is important too. I read an article about a widow in India being giving a loan for a tractor. 2 months payments on the loan have to amount to about a year of her previous income (note that the other 10 months need to be repaid - this is a loan not a gift). A tractor is such a force multiplier that she is able to make the payments all year and save up enough to dream of sending her son to a good school. This whole thing depends on someone being willing to risk giving someone a loan who clearly doesn't make enough to pay it off.
I post this not really as a rebuttal to your post, or the grandparent, but more of a "here's what success a couple of generations later" actually looks like and what some of its characteristics may be.
Luck is important too. I read an article about a widow in India being giving a loan for a tractor. 2 months payments on the loan have to amount to about a year of her previous income (note that the other 10 months need to be repaid - this is a loan not a gift). A tractor is such a force multiplier that she is able to make the payments all year and save up enough to dream of sending her son to a good school. This whole thing depends on someone being willing to risk giving someone a loan who clearly doesn't make enough to pay it off.