  For any number field K with a complex place, we present an infinite family of
elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim
\mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E in
this family, where r_2 is the number of complex places of K. This provides a
counterexample to a conjecture appearing in work of Mazur and Rubin.
