The poem that is usually titled "Harlem" was originally published untitled so it could be more widely applicable to places other than Harlem. The possible scenarios that Hughes lays out are relevant to any dream or dreamer. This universality is one reason for the poem's staying power.
Hughes himself did achieve many of his dreams but he also constantly deferred them. His father encouraged him to attend a predominantly white university in New York, but it was not the right environment for him, as racist attitudes were widespread. Later, at a school of his own choosing, he did earn a college degree.
The success Hughes earned as a writer was associated with that of the vibrant artistic community of the Harlem Renaissance. His decision to become a writer also did not happen overnight. His father opposed this choice. Hughes worked many low-wage jobs--famously in Washington DC as a busboy--while struggling as a writer.
More broadly as well, the poem strikes a chord with Hughes' politics, as a socialist. He was later called to the House of Un-American Activities Committee.