The motion of a sphere accelerating from rest along the center line of a cylindrical tube filled with a polyisobutylene (PIB) Boger fluid is examined both experimentally, using a digital imaging system, and numerically via a Lagrangian finite element method for single and multimode Oldroyd models including inertia. Spheres of diameter 2.54 cm and varying densities are released in a tube of radius 5.2 cm. Deborah numbers in the range 0.4 less-than-or-equal-to De less-than-or-equal-to 11.7 are obtained, and the experimental results show transient velocity overshoots of up to 50% before the spheres approach their final constant settling velocity. Independent measurements of this settling velocity indicate significant drag increases at high De. The numerical calculations are compared with the experimental data, and markedly improved agreement is obtained for multimode simulations.