Nanostructured quantum well and quantum dot solar cells are being widely investigated as a means of extending infrared absorption and enhancing photovoltaic device performance. In this work, we describe the impact of nanostructured layer number on the performance of flexible, highvoltage InGaAs/GaAs quantum well solar cells. Multiple quantum well structures are observed to have a higher short circuit current but a lower open circuit voltage than similar single quantum well structures. Analysis of the underlying dark diode characteristics indicate that these highvoltage structures are limited by radiative recombination at high bias levels. The results of this study suggest that future development efforts should focus on maximizing the current generating capability of a limited number of nanostructured layers and minimizing recombination within the nanostructured absorber.