  The Weak Gravity Conjecture predicts that in quantum gravity there should
exist overcharged states, that is states with charge larger than their mass.
Extending this to large masses and charges, we are expecting similar
overcharged classical solutions. This has been demonstrated in
higher-derivative extensions of General Relativity. In this paper we
investigate the existence of overcharged solutions in General Relativity. We
study the dynamics of a thin shell of mass $m$ and charge $Q$ under the action
of its own gravitational and $U(1)$ fields. We show that shells with surface
energy $\sigma$ and pressure $P$ obeying $P=w\sigma$ with $0\leq w\leq 1$ are
necessarily undercharged $m\geq |Q|$ and always collapse to form
Reissner-Nordstr\"om black holes. Nevertheless, if $-1\leq w<0$, we find that
overcharged $m\leq |Q|$ shells exist, which however, are inevitably stabilized
at finite radial distance.
