Today's perovskite solar cells (PSCs) are limited mainly by their open-circuit voltage (VOC) due to nonradiative recombination. Therefore, a comprehensive understanding of the relevant recombination pathways is needed. Here, intensity-dependent measurements of the quasi-Fermi level splitting (QFLS) and of the VOC on the very same devices, including pin-type PSCs with efficiencies above 20%, are performed. It is found that the QFLS in the perovskite lies significantly below its radiative limit for all intensities but also that the VOC is generally lower than the QFLS, violating one main assumption of the Shockley-Queisser theory. This has far-reaching implications for the applicability of some well-established techniques, which use the VOC as a measure of the carrier densities in the absorber. By performing drift-diffusion simulations, the intensity dependence of the QFLS, the QFLS-VOC offset and the ideality factor are consistently explained by trap-assisted recombination and energetic misalignment at the interfaces. Additionally, it is found that the saturation of the VOC at high intensities is caused by insufficient contact selectivity while heating effects are of minor importance. It is concluded that the analysis of the VOC does not provide reliable conclusions of the recombination pathways and that the knowledge of the QFLS-VOC relation is of great importance.