  In 2000, Croke and Kleiner showed that a CAT(0) group G can admit more than
one boundary. This contrasted with the situation for word hyperbolic groups,
where it was well-known that each such group admitted a unique boundary---in a
very stong sense. Prior to Croke and Kleiner's discovery, it had been observed
by Geoghegan and Bestvina that a weaker sort of uniquness does hold for
boundaries of torsion free CAT(0) groups; in particular, any two such
boundaries always have the same shape. Hence, the boundary really does carry
significant information about the group itself. In an attempt to strengthen the
correspondence between group and boundary, Bestvina asked whether boundaries of
CAT(0) groups are unique up to cell-like equivalence. For the types of space
that arise as boundaries of CAT(0) groups, this is a notion that is weaker than
topological equivalence and stronger than shape equivalence. In this paper we
explore the Bestvina Cell-like Equivalence Question. We describe a
straightforward strategy with the potential for providing a fully general
positive answer. We apply that strategy to a number of test cases and show that
it succeeds---often in unexpectedly interesting ways.
