Buffett has stated on many occasions that he was very lucky: he was born into a fairly rich family in the United States, his father was a stockbroker and he got good genes from his parents. (No chronic health issues, photographic memory, high IQ, etc.) In his own words, he "won the ovarian lottery."
That said, you don't need to have all that to become rich. Others have already mentioned Walter Schloss. He started with absolutely nothing, put all of his (meager) net worth into his hedge fund when he finally put it together, and he managed to become wealthy by following Benjamin Graham's lessons of value investing to the letter.
As someone above already said, don't aim to be the next Buffett - aim to be the best you.