We measured the linear and nonlinear rheology of model polyisoprene comb polymers wit It a moderate number (5-18) of short (marginally entangled to unentangled) branches and highly entangled backbones. The hierarchical modes of relaxation were found to govern both the linear and nonlinear response. Appropriate modification of tube-model theory for entangled branches, inspired by recent work oil asymmetric star polymers (where the short branch behaves as effectively larger oil small time scales), provided a framework for quantitative predictions of the linear viscoelastic spectra, The extended nonlinear stress relaxation data over a wide time range (via time-temperature superposition) obeys time-strain separability and allows extraction of two damping functions one for the branches at short times and one for the diluted backbone at long times. Both exhibit signatures of the comb architecture. The comb damping function at short times, shifted relative to the branch relaxation, is dominated by the retraction of branches and backbone end segments. The backbone damping function is rationalized by considering it as a linear chain that feels it smaller effective strain due to the prior branch relaxation.