The dynamics of dilute solutions of finitely extensible nonlinear elastic (FENE) bead-spring dumbbell and chain models in start-up and relaxation of uniaxial elongational and shear flow and in the flow kinematics obtained from a DNS database of a turbulent channel flow is investigated for Weissenberg numbers (We) of 1, 10, and 100. Five FENE models are considered: FENE chain, FENE dumbbell, FENE-P chain, FENE-P dumbbell, and FENE-PM chain. The FENE chain is used as the standard of reference to evaluate all coarse-grained models. It is shown that in transient elongational flow, pre-averaged models incur large errors while the FENE dumbbell, with appropriate selection of the bead-spring parameters, can provide reasonably accurate predictions once the polymer has unravelled. Faithful prediction of the dynamics prior to polymer unravelling requires a multi-mode model. In transient shear flow, none of the coarse-grained models studied can correctly replicate the dynamics predicted by the FENE chain over the entire range of We considered. The flow in wall-bounded turbulence is shown to be predominantly simple shear within the viscous sublayer and a mixture of simple shear and elongational flow in the buffer layer. The dominant contributions to polymer stress arise from patches of biaxial or uniaxial elongational flow encountered in the buffer layer. The Weissenberg number is the critical parameter which determines polymer stretching, stresses, and drag reduction in a given polymer-solvent system. Effective drag reduction requires a large polymer extensibility and We(tau) similar to O(100) or higher. Comparison of the predictions of different models reveals that the FENE dumbbell, with appropriate selection of the bead-spring parameters, can provide reasonably accurate predictions of the polymer dynamics in turbulent flow at high We(tau). Pre-averaged models, while in qualitative agreement with the FENE chain, are quantitatively inaccurate and over-estimate the polymer stresses by up to 200-400% in regions of strong polymer stretching. These results suggest that the most promising approach to accurate computation of polymer drag reduction at present is through stochastic simulations.