The problem of finding out whether a given integer $D$ may be written as the sum of two rational cubes $x^3+y^3=D$ is one of the oldest problems in number theory. In this paper, we conduct an experimental investigation into the frequency of $D$ which admit two distinct relatively prime solutions. We also evaluate this data against a hypothesis which does afford an explanation for why $D$ which admit three relatively prime solutions are quite rare, and why no $D$ has been found which may be expressed in four different ways as a sum of relatively prime cubes. (From the introduction)