Accurate models of the star formation histories (SFHs) of recently quenched galaxies can provide constraints on when and how galaxies shut down their star formation. The recent development of nonparametric SFH models promises the flexibility required to make these measurements. However, model and prior choices significantly affect derived SFHs, particularly for post-starburst galaxies (PSBs), which have sharp changes in their recent SFH. In this paper, we create mock PSBs, then use the Prospector SED fitting software to test how well four different SFH models recover key properties. We find that a two-component parametric model performs well for our simple mock galaxies, but is sensitive to model mismatches. The fixed- and flexible-bin nonparametric models included in Prospector are able to rapidly quench a major burst of star formation, but systematically underestimate the post-burst age by up to 200 Myr. We develop a custom SFH model that allows for additional flexibility in the recent SFH. Our flexible nonparametric model is able to constrain post-burst ages with no significant offset and just similar to 90 Myr of scatter. Our results suggest that while standard nonparametric models are able to recover first-order quantities of the SFH (mass, SFR, average age), accurately recovering higher-order quantities (burst fraction, quenching time) requires careful consideration of model flexibility. These mock recovery tests are a critical part of future SFH studies. Finally, we show that our new, public SFH model is able to accurately recover the properties of mock star-forming and quiescent galaxies and is suitable for broader use in the SED fitting community. https://github.com/bd-j/prospector