The dominant feedback mechanism in low-mass haloes is usually assumed to take the form of massive stars exploding as supernovae (SNe). We perform very high resolution cosmological zoom-in simulations of five dwarf galaxies to z = 4 with our mechanical SN feedback model. This delivers the correct amount of momentum corresponding to the stage of the SN remnant evolution resolved, and has been shown to lead to realistic dwarf properties in isolated simulations. We find that in four out of our five simulated cosmological dwarfs, SN feedback has insufficient impact resulting in excessive stellar masses, extremely compact sizes and central supersolar stellar metallicities. The failure of the SN feedback in our dwarfs is physical in nature within our model and is the result of the build-up of very dense gas in the early universe due to mergers and cosmic inflows prior to the first SN occurring. We demonstrate that our results are insensitive to resolution (provided that it is high enough), details of the (spatially uniform) UV background and reasonable alterations within our star formation prescription. We therefore conclude that the ability of SNe to regulate dwarf galaxy properties is dependent on other physical processes, such as turbulent pressure support, clustering, and runaway of SN progenitors and other sources of stellar feedback.