Non-radiative electron-hole recombination is the bottleneck to efficient kesterite thin-film solar cells. We have performed a search for active point-defect recombination centres using first-principles calculations. We show that the anion vacancy in Cu2ZnSnS4 (CZTS) is electrically benign without a donor level in the bandgap. V_S can still act as an efficient non-radiative site through the aid of an intermediate excited state involving electron capture by Sn. The bipolaron associated with Sn 4+ to Sn 2+ two-electron reduction stabilizes the neutral sulfur vacancy over the charged states; however, we demonstrate a mechanism whereby non-radiative recombination can occur via multiphonon emission. Our study highlights that defect-mediated recombination does not require a charge transition level deep in the bandgap of a semiconductor. We further identify Sn_Zn as the origin of persistent electron trapping/detrapping in kesterite photovoltaic devices, which is suppressed in the selenide compound.