The observed number of dwarf galaxies as a function of rotation velocity is significantly smaller than predicted by the standard model of cosmology. This discrepancy cannot be simply solved by assuming strong baryonic feedback processes, since they would violate the observed relation between maximum circular velocity (vmax) and baryon mass of galaxies. A speculative but tantalizing possibility is that the mismatch between observation and theory points towards the existence of non-cold or non-collisionless dark matter (DM). In this paper, we investigate the effects of warm (WDM), mixed (MDM, i.e. warm plus cold), and self-interacting DM (SIDM) scenarios on the abundance of dwarf galaxies and the relation between observed H i line width and maximum circular velocity. Both effects have the potential to alleviate the apparent mismatch between the observed and theoretical abundance of galaxies as a function of vmax. For the case of WDM and MDM, we show that the discrepancy disappears, even for lukewarm models that evade stringent bounds from the Lyman-α forest. SIDM scenarios can also provide a solution as long as they lead to extended (≳1.5 kpc) DM cores in the density profiles of dwarf galaxies. Only models with velocity-dependent cross-sections can yield such cores without violating other observational constraints at larger scales.