We investigate the influence of feedback on the capacity of multiple-input multiple-output (MIMO) non-coherent fading channels with memory. We derive an upper bound on the feedback capacity of regular fading channels, which is shown to coincide with a previously known upper bound on the capacity without feedback. Hence, it is concluded that whereas feedback does in general increase capacity, this increase is relatively small in the sense that the same upper bound holds for both scenarios. From this bound we derive an upper bound on the MIMO fading number in the presence of feedback and show that in the single-input single-output (SISO) case this bound is tight, i.e., the SISO fading number is not changed by feedback. Next, we derive an upper bound on the feedback capacity of spatially independent MIMO Gaussian fading channels. From this bound we derive a new upper bound on the corresponding fading number and show that in the multiple-input single-output (MISO) case this bound is tight. Finally, in the case of a non-regular SISO Gaussian fading process we show that feedback does not increase the pre-log, i.e., the ratio of the capacity to the logarithm of the signal-to-noise ratio log SNR is not changed in the limit when the SNR tends to infinity.