Let C be any family of 2n disjoint compact convex sets in the plane. If C has f-equal width for some direction f, then 2n/3 pairs of elements in C can always be matched by disjoint line segments and more than 4n/5 pairs cannot occasionally be matched. Furthermore, if C is a family of translates of the set satisfying a certain constraint, then 4n/5 pairs of elements can always be matched by disjoint line segments.