The motion of a viscoelastic liquid, modeled via the Chilcott–Rallison version of the FENE dumbbell model, is studied numerically for a co‐rotating two‐roll mill. Solutions are obtained for values of the Deborah number up to 13.6, and for the dimensionless contour length 300≤L2≤2000 at three different values of the concentration parameter c≤1. The emphasis of the study is on start‐up flow from a state of rest. Of particular interest is the onset of a complicated overshoot and undershoot of the dumbbell configuration at very large values of De and L. This phenomenon is due primarily to the fact that the flow is significantly modified as the polymer contribution to the stress increases, but there is also a contribution due to the closed nature of the flow system which means that all streamlines of the flow are closed, and the deformation history experienced by a polymer molecule is therefore periodic in time.