A new method for refining the predictions of a finitely extensible bead-spring chain model with hydrodynamic interactions is introduced. The suggested successive fine-graining procedure involves the representation of a long but finite macromolecule of N-K Kuhn steps with a coarse-grained bead-spring chain model, and subsequently "fine graining" the representation by progressively increasing the number of beads, N. Extrapolating the results obtained using exact Brownian dynamics simulations for several values of N to the limit N-1-->N-K, leads to improved estimates for the behavior of the actual polymer chains. The extrapolated results for the extensional viscosity are found to be in good agreement with the recent experimental observations of Gupta et al. (2000). The scaling of the extensional viscosity with molecular weight is also examined. (C) 2004 The Society of Rheology.