A novel tolerance ring includes a hollow cylindrical body defining a longitudinal axis that passes axially through its center. A circumferential gap divides the hollow cylindrical body so that the hollow cylindrical body includes first and second longitudinal edges that define the circumferential gap. A plurality of bumps protrude radially from the hollow cylindrical body, and consists of a plurality of gap-adjacent bumps and a plurality of non-gap-adjacent bumps. At least one of the plurality of bumps is circumferentially between the gap and each of the non-gap-adjacent bumps. None of the plurality of bumps is circumferentially between the gap and each of the gap-adjacent bumps. The tolerance ring includes a means for increasing an otherwise lower average radial compressive stiffness of the gap-adjacent bumps to become substantially equal to an average radial stiffness of the non-gap-adjacent bumps.