It is wise to start with definitions. Most people will say that the American Dream is synonymous with the ideals of freedom, equality, the reward for hard work, and material and personal comfort.
If this is the byline for the American dream, then the main players in the book have it. Jay Gatsby is rich beyond measure and has comfort in excess. Tom, Daisy, and Jordan, the old moneyed people, are also fabulously rich and comfortable. In this sense, they are proof of the American Dream.
However, as we look at their lives, they are far from happy, or even decent people. Fitzgerald's point is that the American dream is not what it is cracked up to be. Money, wealth, and great parties do not create joy. Gatsby, in this sense, is tragic, as are the others. In some way, the others are far worse. Tom and Daisy have affairs (both not happy) and Jordan is a cheat.
One way to interpret Fitzgerald is to say that his vision of the American Dream as most of the people in 1922 saw it was hollow.
Back in the early 20th century (and maybe even today), the American Dream was to make money. People immigrated from all over the world to the United States in the hopes of living a better life.
In The Great Gatsby, Fitzgerald manages to define, praise, and condemn the idea of the American Dream. He is able to do this by separating The East Egg (New money) from The West Egg (Old Money). Old Money was able to do anything; they had the connections as well as wealth. New Money recently came about and they only had the wealth. So, Gatsby desperately wanted to be Daisy, who was Old Money, but he could not. This goes to show Fitzgerald's idea that people of the same class stick together (even after Daisy learns about Tom's affair, they later eat chicken together).
I feel that Fitzgerald has a very accurate view of the world. The old proverb "Money doesn't buy happiness" definitely comes about in this novel. Gatsby has everything he could possibly imagine, except for the love of his life, Daisy. Everyone leads tragic lives because of their greed for money.