By combining rheology, light scattering, small angle X-ray scattering and simulations we explore the glassy dynamics of soft colloids using microgels and charged particles interacting by steric and screened Coulomb interactions, respectively. In the supercooled regime, the structural relaxation time (SRT) of both systems grows steeply with volume fraction, reminiscent of the behavior of colloidal hard sphere systems [1]. This suggests that softness has no impact on the growth of the SRT on approaching the glass transition, as confirmed by computer simulations. We show that the onset of a finite yield-stress, occurring within the supercooled regime, systematically corresponds to a minimum of the stretching exponent characterizing the decay of the two-time intensity autocorrelation function. Softness becomes relevant only at very large packing fractions, when the system falls out of equilibrium. In this non-equilibrium regime, the SRT depends surprisingly weakly on packing fraction and time correlation functions exhibit compressed exponential decays consistent with internal stress-driven relaxation. The transition to this novel regime coincides with the onset of an anomalous decrease of local order with increasing density, reminiscent of the reentrant glass transition predicted theoretically in ultrasoft systems [2,3], and corresponds to a weakening of the volume fraction dependence of the plateau modulus of the suspensions. We propose that these peculiar dynamics result from a competition between the non-equilibrium aging dynamics of the glassy state and the tendency of soft systems to refluidize at high packing fractions.