  Certain four-dimensional $N=4$ supersymmetric theories have special vacua in
which massive charged vector supermultiplets become massless, resulting in an
enhanced non-abelian gauge symmetry. We show here that any two $N=4$ theories
having the same Bogomolnyi spectrum at corresponding points of their moduli
spaces have the same enhanced symmetry groups. In particular, the $K_3\times
T^2$ compactified type II string is argued to have the same enhanced symmetry
groups as the $T^6$-compactified heterotic string, giving further evidence for
our conjecture that these two string theories are equivalent. A feature of the
enhanced symmetry phase is that for every electrically charged state whose mass
tends to zero as an enhanced symmetry point is approached, there are
magnetically charged and dyonic states whose masses also tend to zero, a result
that applies equally to N=4 super Yang-Mills theory. These extra
non-perturbative massless states in the $K_3$ compactification result from
$p$-branes wrapping around collapsed homology two-cycles of $K_3$. Finally, we
show how membrane `wrapping modes' lead to symmetry enhancement in D=11
supergravity, providing further evidence that the $K_3$-compactified D=11
supergravity is the effective field theory of the strong coupling limit of the
$T^3$-compactified heterotic string.
