NGC 6388 and NGC 6441 are two massive Galactic bulge globular clusters that share many properties, including the presence of an extended horizontal branch (HB), quite unexpected because of their high metal content. In this paper we use Hubble Space Telescope's WFPC2, ACS, and WFC3 images and present a broad multicolor study of their stellar content, covering all main evolutionary branches. The color-magnitude diagrams (CMDs) give compelling evidence that both clusters host at least two stellar populations, which manifest themselves in different ways. NGC 6388 has a broadened main sequence (MS), a split sub-giant branch (SGB), and a split red giant branch (RGB) that becomes evident above the HB in our data set; its red HB is also split into two branches. NGC 6441 has a split MS, but only an indication of two SGB populations, while the RGB clearly splits in two from the SGB level upward, and no red HB structure. The multicolor analysis of the CMDs confirms that the He difference between the two main stellar populations in the two clusters must be similar. This is observationally supported by the HB morphology, but also confirmed by the color distribution of the stars in the MS optical band CMDs. However, a MS split becomes evident in NGC 6441 using UV colors, but not in NGC 6388, indicating that the chemical patterns of the different populations are different in the two clusters, with C, N, and O abundance differences likely playing a major role. We also analyze the radial distribution of the two populations. Based on proprietary and archival observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.