For any closed surface $S$ of genus $g \geq 2$, we show that the deformation
space of marked hyperbolic 3-manifolds homotopy equivalent to $S$, $AH(S \times
I)$, is not locally connected. This proves a conjecture of Bromberg who
recently proved that the space of Kleinian punctured torus groups is not
locally connected. Playing an essential role in our proof is a new version of
the filling theorem that is based on the theory of cone-manifold deformations
developed by Hodgson, Kerckhoff, and Bromberg.