Planetary systems with multiple transiting planets present natural laboratories for testing planet formation theories. We characterize such a system, the bright (V=8.97) Sun-like star HD 191939 (TOI 1339) which hosts three transiting sub-Neptunes first discovered by TESS. In our RV time series of 100+ observations from Keck/HIRES and the APF, we find a non-transiting fourth planet and a fifth long-period companion. We measure the masses of all four planets and explore the dynamics of the system. We find the likelihood of planets c and d to be in mean-motion 4:3 resonance at ~50%. This system is ideal for further atmospheric follow up as it is bright and contains relatively low-mass planets.