The open-circuit voltages (V OC) of hybrid perovskite (HP) solar cells do not increase sufficiently with increasing bandgap (for Eg > 1.70eV). We study the impact of A+ size mismatch induced lattice distortions (in ABX3 structure) on the optoelectronic quality of high-bandgap HPs and find that the highest quality films have high A-site size-mismatch, where large guanidinium (GA) compensates for small Cs to keep the tolerance factor in the range for the perovskite structure. Specifically, we find that 1.84eV bandgap (FA0.33GA0.19Cs0.47)­Pb­(I0.66Br0.34)3 and 1.75eV bandgap (FA0.58GA0.10Cs0.32)­Pb­(I0.73Br0.27)3 attain quasi-Fermi level splitting of 1.43eV and 1.35eV, respectively, which is >91% of the Shockley-Queisser limit for both cases. Films of 1.75eV bandgap (FA,GA,Cs)­Pb­(I,Br)3 are then used to fabricate p-i-n photovoltaic devices that have a V OC of 1.24 V. This V OC is among the highest V OC reported for any HPs with similar bandgap (1.7 to 1.8 eV) and a substantial improvement for the p-i-n architecture, which is desirable for tandems with Si, CIGS, or a low-bandgap HP. Collectively, our results show that non-radiative recombination rates are reduced in (FA,GA,Cs)­Pb­(I,Br)3 films and prove that FA-GA-Cs alloying is a viable route to attain high V OC in high-bandgap HP solar cells.