We present the first quantitative experimental measurements of the transient motion of a sphere as it accelerates from rest along the centerline of a tube containing a highly elastic, shear-thinning, aqueous polyacrylamide solution. For all shear-rate-dependent Deborah numbers (1.6 less than or equal to De(gamma) less than or equal to 4.2) and sphere-to-tube ratios (0.089, less than or equal to a/R less than or equal to 0.387) investigated, transient oscillations in the velocity of the sphere are observed, often causing the sphere to "rebound," or reverse directions during the first oscillation. These measurements are in qualitative agreement with the analysis of King and Waters (1972) who presented an analytic solution for the transient motion of a sphere through an unbounded domain of fluid described by the linear Jeffreys model. We also show a similar response in one-dimensional creep experiments which are described quantitatively by a multimode formulation of the upper-convected Maxwell model including a solvent retardation term. Such experiments isolate the shear kinematics from the combined shear and extensional flow around a sphere and indicate that because of the slow quadratic growth of the elastic normal stresses in the fluid at short times and small fluid strains, the initial transient motion of the sphere is governed primarily by a balance of linear viscoelastic stresses and the inertia of the sphere.