Large extensional deformations of viscoelastic fluid columns in filament stretching rheometers are studied through numerical simulations up to Hencky strains of greater than epsilon = 4. The time-dependent axisymmetric calculations incorporate the effects of viscoelasticity, surface tension, fluid inertia, plus a deformable foe surface and provide quantitative descriptions of the evolution in the filament profile, the kinematics in the liquid column and the resulting dynamic evolution in the viscous and elastic contributions to the total stress. In addition to investigating the variation in the apparent Trouton ratio expected in experimental measurements using this new type of extensional rheometer, we also investigate the generic differences between the response of Newtonian and viscoelastic fluid filaments described by the Oldroyd-B model. For small strains, the fluid deformation is governed by the Newtonian solvent contribution to the stress and the filament evolution is very similar in both the Newtonian and viscoelastic cases. However, in the latter case at large strains and moderate Deborah numbers, elastic stresses dominate leading to strain-hardening in the axial mid-regions of the column and subsequent drainage of the quasi-static liquid reservoir that forms near both end-plates. These observations are in good qualitative agreement with experimental observations. For small initial aspect ratios and low strains, the non-homogeneous deformation predicted by numerical simulations is well described by a lubrication theory solution. At larger strains, the initial flow non-homogeneity leads to the growth of viscoelastic stress boundary layers near the free surface which can significantly affect the transient Trouton ratio measured in the device. Exploratory design calculations suggest that mechanical methods for modifying the boundary conditions at the rigid end-plates can reduce this non-homogeneity and lead to almost ideal uniaxial elongational Bow kinematics.