Personally, I think the American dream applies to those who come from [extreme] poverty. It is alive and well for those who come from literally next to nothing, work hard and have something to show for it, raise their kids in a place where they can get a good education, and enjoy their family.
But the American Dream also suggests that it is for the generation of the person working to achieve it. For most people, it's a process that yields dividends a generation or two later. Being able to achieve the American Dream in one lifetime isn't impossible, but it is extremely difficult.
I think a lot of people think American Dream and picture Scarface. It's subjective and depends on what your definition of "making it" is. Being poor as hell on food stamps, living in nothing but apartments your entire life, to going out and earning a degree, buying a home in a good area and taking care of your family sounds like someone living the American dream to me. If you can't be happy with that progress in your life, you need to learn to be content. Some people chase fortune their entire lives, get it, and they are still miserable.