The cold dark matter halo mass function is much steeper than the galaxy stellar mass function on galactic and subgalactic scales. This difference is usually reconciled by assuming that the galaxy formation efficiency drops sharply with decreasing halo mass, so that virtually no dwarf galaxies form in haloes less massive than ∼1010 M⊙. In turn, this implies that, at any given radius, the dark mass enclosed by a galaxy must exceed a certain minimum. We use rotation curves of dwarf galaxies compiled from the literature to explore whether their enclosed mass is consistent with these constraints. We find that almost one-half of the dwarfs in our sample with stellar mass in the range of 106 < Mgal/M⊙ < 107 are at odds with this restriction: either they live in haloes with masses substantially below 1010 M⊙ or there is a mechanism capable of reducing the dark mass enclosed by some of the faintest dwarfs. Neither possibility is easily accommodated within the standard Λ cold dark matter scenario. Extending galaxy formation to haloes well below 1010 M⊙ would lead to large numbers of dwarf galaxies in excess of current estimates; at the same time, the extremely low stellar mass of the systems involved makes it unlikely that baryonic effects can reduce their dark matter content. Resolving this challenge seems to require new insights into dwarf galaxy formation, or perhaps a radical revision of the prevailing paradigm.