We report the detection and analysis of the red giant branch (RGB) luminosity function bump in a sample of isolated dwarf galaxies in the Local Group. We have designed a new analysis approach comparing the observed color-magnitude diagrams (CMDs) with theoretical best-fit CMDs derived from precise estimates of the star formation histories of each galaxy. This analysis is based on studying the difference between the V magnitude of the RGB bump and the horizontal branch at the level of the RR Lyrae instability strip (ΔV bump HB ) and we discuss here a technique for reliably measuring this quantity in complex stellar systems. By using this approach, we find that the difference between the observed and predicted values of ΔV bump HB is +0.13 ± 0.14 mag. This is smaller, by about a factor of 2, than the well-known discrepancy between theory and observation at low metallicity commonly derived for Galactic globular clusters (GCs). This result is confirmed by a comparison between the adopted theoretical framework and empirical estimates of the ΔV bump HB parameter for both a large database of Galactic GCs and for four other dwarf spheroidal galaxies for which this estimate is available in the literature. We also investigate the strength of the RGB bump feature (R bump), and find very good agreement between the observed and theoretically predicted R bump values. This agreement supports the reliability of the evolutionary lifetimes predicted by theoretical models of the evolution of low-mass stars. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. These observations are associated with program 10505.