Yes, you can become rich if you're born poor. Just realize you have the deck stacked against you from the day you're born all the way into adulthood. Doesn't make it impossible to rise in society, it's just a bit harder.
At the same time, the rich enjoy numerous advantages: private schools, private tutoring, a valuable social network of relatives and parents, access to highly qualified doctors/psychologists when needed and lots of other benefits. Access to better nutrition, less parental violence and drug abuse, a household stacked with scientific books rather than comic books.
If you are born rich (social studies call it 'high socio-economic status') you are way more likely to gain a highly paid job or run a successful company than if you are born poor. In other words: controlling for graduate degree GPAs or IQ, members of the upper class still do better economically by a full standard deviation than their middle class competitors. Which is massive.