Let R be a commutative noetherian local ring, and M a finitely generated R-module of infinite projective dimension. It is known that the depths of the syzygy modules of M eventually stabilize to the depth of R. We will consider conditions under which a similar statement can be made regarding dimension. In particular, we show that if R is equidimensional and the Betti numbers of M are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of M coincides with the dimension of R. This is joint work with K. Beck at the University of Arizona.