Values of the ratio of the longitudinal diffusion coefficient to mobility D(L)/mu for electrons in He, Ar, Kr, and Xe are derived from current waveforms obtained during earlier measurements of electron mobility. The electric field to gas density ratios E/N cover the wide range of 10(-3) to 20 Td, thereby bridging previous experiments at low E/N to recent experiments at high E/N. Here 1 Td = 1 x 10(-21) V m2. The corresponding D(L)/mu values range from 0.0066 eV for thermal electrons at 77 K to 10 eV. In addition to the well-known peak in D(L)/mu for Ar at E/N between 0.01 and 0.1 Td caused by the Ramsauer minimum in the momentum transfer cross section, we find previously unreported low-energy peaks in D(L)/mu vs E/N in Kr and Xe and previously unreported pronounced leveling-off in D(L)/mu at E/N > 8 Td in Ar, Kr, and Xe. Calculations of transport coefficients using numerical solutions of the Boltzmann equation and cross section sets in the literature give good agreement with experiment from E/N producing thermal electrons up to average energies almost-equal-to 10 eV and E/N up to 100 Td, the upper limit of our calculations. The leveling off of D(L)/mu at high E/N is caused by inelastic collisions.