In this paper we consider the question of how many possible dimensions a basin boundary can have. We conjecture that the number of possible dimension values is at most the number of some well defined asymptotic sets (called basic sets) on the basin boundary. It should be noticed that the dimension of a basic set also has a dynamical meaning. The conjecture will be proved for a class of Axiom A systems (namely two dimensional diffeomorphisms and one dimensional chaotic maps). In addition, we will give numerical evidence for a physical example.