  With the question, ``Can relativistic charged spheres form extremal black
holes?" in mind, we investigate the properties of such spheres from a classical
point of view. The investigation is carried out numerically by integrating the
Oppenheimer-Volkov equation for relativistic charged fluid spheres and finding
interior Reissner-Nordstr\"om solutions for these objects. We consider both
constant density and adiabatic equations of state, as well as several possible
charge distributions, and examine stability by both a normal mode and an energy
analysis. In all cases, the stability limit for these spheres lies between the
extremal ($Q = M$) limit and the black hole limit ($R = R_+$). That is, we find
that charged spheres undergo gravitational collapse before they reach $Q = M$,
suggesting that extremal Reissner-Nordtr\"om black holes produced by collapse
are ruled out. A general proof of this statement would support a strong form of
the cosmic censorship hypothesis, excluding not only stable naked
singularities, but stable extremal black holes. The numerical results also
indicate that although the interior mass-energy $m(R)$ obeys the usual $m/R <
4/9$ stability limit for the Schwarzschild interior solution, the gravitational
mass $M$ does not. Indeed, the stability limit approaches $R_+$ as $Q \to M$.
In the Appendix we also argue that Hawking radiation will not lead to an
extremal Reissner-Nordstr\"om black hole. All our results are consistent with
the third law of black hole dynamics, as currently understood.
