This note addresses the problem of stabilizing a multi-input multi-output (MIMO) discrete-time linear time invariant (LTI) system over a MIMO additive noise channel. To treat this problem, we assume that the communication link between the plant output and the controller input consists of multiple additive colored mutually correlated noise channels subject to independent signal-to-noise ratio (SNR) constraints. We derive analytical conditions for which mean square stabilization (MSS) can be achieved under such constraints. We also formulate numerical methods to test these conditions when the noise is white and correlated. Moreover, for simpler plant models, a characterization of the set of power constraints compatible with MSS is obtained. Our results show that the frequency response of the spectral factor related to the channel noise affects the minimum SNR for stability depending mostly on the unstable poles and their directions. This is aggravated by the existence of nonminimum phase zeros and higher relative degree of the plant. On the other hand, we detect that systems under highly correlated noise show lower SNR requirements for stability compared to ones with independent noise channels. Numerical simulations are provided to illustrate the theoretical results.