Several donor-acceptor polymers have achieved power conversion efficiency (PCE) ≈10% and internal quantum efficiency (IQE) >90% in optimized polymer-fullerene bulk heterojunction solar cells. However, most of these devices optimize with active layers <200-nm-thick and the devices suffer from poor fill factors (FF) when the active layer thickness is increased to improve light absorption. To increase the PCE of bulk heterojunction solar cells beyond 15%, 300-nm-thick devices with FF ≈0.8 and external quantum efficiency (EQE) >90% are needed. We use a combination of experimental results and a numerical device simulator to investigate recombination and charge-carrier transport in optically thick BHJ solar cells. To validate the use of the device simulator, we fabricate a large variety of BHJ solar cells with hole mobility ranging from 1.6x10-7 to 3.6x10-4 cm2 V-1 s-1 and active layer thickness ranging from 60 to 350 nm. We reproduced the wide range of experimental device results with the device simulator using only two fit parameters and the experimentally measured electron and hole mobility. With this approach, we determine that electron and hole mobility must be >4x10-3 cm2 Vs-1 in order to attain a 0.8 FF in a 300-nm-thick device with recombination rate constant similar to P3HT:PCBM. This result is promising because the electron mobility of the most commonly used fullerene-derivative, PCBM, is already 5x10-3 cm2 Vs-1. Thus, the hole mobility of donor polymers only needs to be increased from the typical values of 10-4 to 4x10-3 cm2 Vs-1 in order to achieve a single junction bulk heterojunction solar cell with EQE >90% and FF =0.8. We also examine the effect of the recombination rate constant on device FF and find that decreasing the recombination rate constant of BHJ solar cells would significantly improve device performance. Our results suggest that researchers should prioritize improving charge-carrier mobility when synthesizing the next generation of semiconducting materials for BHJ solar cells and highlight that researchers should aim to understand what factors affect the recombination rate constant in these devices. Moreover, our findings show that device simulators provide valuable insight into BHJ solar cell operation and can be used to rapidly determine how certain variables affect device performance.