Population density is not really a problem: Most of the country is virtually empty, but there are a series of urban agglomerations that have incredibly high density overall. The North-East corridor (DC to Boston, more or less) is the most obvious one, but Chicago and environment or Coastal California are great options, too.

The "secret" reason why it's not happening is public indifference and corporate sabotage. A campaign of decades of worsening public transportation has made people convinced that a high speed train would be just for poor people, which they imagine to be someone else. Also, eminent domain land seizures are slow, environmental impact studies slower, and both force costly changes from original plan that the public hears about as cost-overruns.

Final nail in the coffin: in America, for bizarre reasons, passenger rail has lower priority than freight rail. The freight rail companies don't want to give up the privilege, and obviously you can't have a high speed service wait on freight trains bumbling by.