New non-asymptotic upper bounds on the capacity of non-coherent multiple-input multiple-output (MIMO) Gaussian fading channels with memory are proposed. These bounds are used to derive upper bounds on the fading number of regular Gaussian fading channels and on the pre-log of non-regular ones. The resulting bounds are tight in the multiple-input single-output (MISO) spatially IID Gaussian case whence they yield the exact fading number and pre-log. A new approach is proposed for the derivation of lower bounds on the fading number of MIMO channels. This approach is applied to derive a lower bound on the fading number of spatially IID zero-mean Gaussian fading channels with memory. The new upper and lower bounds on the fading number demonstrate that when the number of receive antennas does not exceed the number of transmit antennas, the fading number of zero-mean spatially IID slowly varying Gaussian MIMO channels is proportional to the number of degrees of freedom, i.e., to the minimum of the number of transmit and receive antennas. We conjecture that the same is true also when the number of receive antennas exceeds the number of transmit antennas. The single-input multiple-output case that was recently solved by Lapidoth & Moser supports this conjecture.