This study reports the results of a systematic numerical investigation, using the upper-convected Maxwell (UCM) and Phan-Thien-Tanner(PTT) models, of viscoelastic fluid flow through three-dimensional gradual planar contractions of various contraction ratios with the aim of investigating experimental observations of extremely large near-wall velocity overshoots in similar geometries [R.J. Poole, M.P. Escudier, P.J. Oliveira, Laminar flow of a viscoelastic shear-thinning liquid through a plane sudden expansion preceded by a gradual contraction, Proc. Roy. Soc. Lond. Set. A 461 (2005) 3827]. We are able to obtain good qualitative agreement with the experiments, even using the UCM model in creeping-flow conditions, showing that neither inertia, second normal-stress difference nor shear-thinning effects are required for the phenomenon to be observed. Guided by the numerical results we propose a simple explanation for the occurrence of the velocity overshoots and the conditions under which they arise. (C) 2009 Elsevier B.V. All rights reserved.