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 analyse 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 theory, 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 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 behaviour of the corresponding vector models. Separately, we show that the 'partial masslessness' exhibited by (plain) massive spin-2 models in de Sitter backgrounds is shared by the three-term system: it also enjoys a reduced local gauge invariance when this mass parameter is tuned to the cosmological constant.