  We consider gedanken experiments to destroy an extremal or nearly extremal
Kerr-Newman black hole by causing it to absorb matter with sufficient charge
and/or angular momentum as compared with energy that it cannot remain a black
hole. It was previously shown by one of us that such gedanken experiments
cannot succeed for test particle matter entering an extremal Kerr-Newman black
hole. We generalize this result here to arbitrary matter entering an extremal
Kerr-Newman black hole, provided only that the non-electromagnetic contribution
to the stress-energy tensor of the matter satisfies the null energy condition.
We then analyze the gedanken experiments proposed by Hubeny and others to
over-charge and/or over-spin an initially slightly non-extremal Kerr-Newman
black hole. Analysis of such gedanken experiments requires that we calculate
all effects on the final mass of the black hole that are second-order in the
charge and angular momentum carried into the black hole, including all
self-force effects. We obtain a general formula for the full second order
correction to mass, $\delta^2 M$, which allows us to prove that no gedanken
experiments of the generalized Hubeny type can ever succeed in over-charging
and/or over-spinning a Kerr-Newman black hole, provided only that the
non-electromagnetic stress-energy tensor satisfies the null energy condition.
Our analysis is based upon Lagrangian methods, and our formula for the
second-order correction to mass is obtained by generalizing the canonical
energy analysis of Hollands and Wald to the Einstein-Maxwell case. Remarkably,
we obtain our formula for $\delta^2 M$ without having to explicitly compute
self-force or finite size effects. Indeed, in an appendix, we show explicitly
that our formula incorporates both the self-force and finite size effects for
the special case of a charged body slowly lowered into an uncharged black hole.
