  In three dimensions, there are two distinct mass-generating mechanisms for
gauge fields: adding the usual Proca/Pauli-Fierz, or the more esoteric
Chern-Simons (CS), terms. Here we analyze the three-term models where both
types are present, and their various limits. Surprisingly, in the tensor case,
these seemingly innocuous systems are physically unacceptable. If the sign of
the Einstein term is ``wrong'' as is in fact required in the CS case, then the
excitation masses are always complex; with the usual sign, there is a (known)
region of the two mass parameters where reality is restored, but instead we
show that a ghost problem arises, while, for the ``pure mass'' two-term system
without an Einstein action, complex masses are unavoidable. This contrasts with
the smooth behavior of the corresponding vector models. Separately, we show
that the ``partial masslessness'' exhibited by (plain) massive spin-2 models in
de Sitter backgrounds is formally shared by the three-term system: it also
enjoys a reduced local gauge invariance when this mass parameter is tuned to
the cosmological constant.
