We present a new observable?Rbump?which is the ratio between the star counts across the red giant branch (RGB) bump and fainter RGB stars to investigate the occurrence of a deep-mixing phenomenon during these evolutionary phases. The comparison between predicted and empirical Rbump-values, based on a large and homogeneous set of Hubble Space Telescope data, brings out that evolutionary lifetimes predicted by canonical RGB models do account for the bulk of Galactic globular clusters included in our sample (29). This evidence suggests that bump and fainter RGB stars do not show the occurrence of deep mixing, which significantly changes their chemical stratification. A few possible exceptions to this general rule are briefly discussed.