Improving devices incorporating solution-processed nanocrystal-based semiconductors requires a better understanding of charge transport in these complex, inorganic-organic materials. Here we perform a systematic study on PbS nanocrystal-based diodes using temperature-dependent current-voltage characterization and thermal admittance spectroscopy to develop a model for charge transport that is applicable to different nanocrystal-solids and device architectures. Our analysis confirms that charge transport occurs in states that derive from the quantum-confined electronic levels of the individual nanocrystals and is governed by diffusion-controlled trap-assisted recombination. The current is limited not by the Schottky effect, but by Fermi-level pinning because of trap states that is independent of the electrode-nanocrystal interface. Our model successfully explains the non-trivial trends in charge transport as a function of nanocrystal size and the origins of the trade-offs facing the optimization of nanocrystal-based solar cells. We use the insights from our charge transport model to formulate design guidelines for engineering higher-performance nanocrystal-based devices.