My expertise is in the history of New York, and from what I have come across, most of the very wealthy people in early New York were already wealthy before they got here, or at the very least prominent citizens. This would include families like the Van Rensselaers, the Livingstons, and the Van Cortlandts. However, that's not a hard and fast rule--the Vanderbilts were middle class when they came over, and so probably was the first Roosevelt (though so little of him is known, it's hard to say for sure).

Going from lower class to middle class (and not outright "riches"), on the other hand, has been a pretty common occurrence since the beginning of the European settlement. In New York, many early colonists came over as indentured servants or otherwise owing a debt for their passage over and renting land. Once they'd worked off the debt, they could buy farms of their own and move up to a more middle class life.

That all said, the American "rags to riches" story was less the result of real life examples, and more the result of the popularity of late 19th Century children's author Horatio Alger. He wrote dozens of books featuring poor children, often orphans and often urban, who rose up to become financially secure if not outright rich.

While Alger's name might not be universally recognized today, it once was. He was one of if not the best-selling author in the United States between the 1870s and his death in 1899.

Because of his reliance on the "rags to riches" narrative, it is often dismissed as the "Horatio Alger myth" and has been criticized by social commentators, scholars, and other authors, such as in this article.