Single formamidinium lead bromide (FAPbBr(3)) perovskite nanocubes, approximately 10 nm in size, have extinction cross sections orders of magnitude larger than single dye molecules and can therefore be used to photoexcite one single dye molecule within their immediate vicinity by means of excitation-energy transfer (EET). The rate of photon emission by the single dye molecule is increased by 2 orders of magnitude under excitation by EET compared to direct excitation at the same laser fluence. Because the dye cannot accommodate biexcitons, NC biexcitons are filtered out by EET, giving rise to up to an order-of-magnitude improvement in the fidelity of photon antibunching. We demonstrate here that, contrary to expectation, energy transfer from the nanocrystal to dye molecules does not depend on the spectral line widths of the donor and acceptor and is therefore not governed by Forster's theory of resonance energy transfer (FRET). Two different cyanine dye acceptors with substantially different spectral overlaps with the nanocrystal donor show a similar light-harvesting capability. Cooling the sample from room temperature to 5 K reduces the average transition line widths 25-fold but has no apparent effect on the number of molecules emitting, i.e., on the spatial density of single dye molecules being photoexcited by single nanocrystals. Narrow zero-phonon lines are identified for both donor and acceptor, with an energetic separation of over 40 times the line width, implying a complete absence of spectral overlap-even though EET is evident. Both donor and acceptor exhibit spectral fluctuations, but no correlation is apparent between the jitter, which controls spectral overlap, and the overall light harvesting. We conclude that the energy transfer process is fundamentally nonresonant, implying effective energy dissipation in the perovskite donor because of strong electron-phonon coupling of the carriers comprising the exciton. The work highlights the importance of performing cryogenic spectroscopy to reveal the underlying mechanisms of energy transfer in complex donor-acceptor systems.