  We consider dimensionally reduced three-dimensional supersymmetric
Yang-Mills-Chern-Simons theory. Although the N=1 supersymmetry of this theory
does not allow true massive Bogomol'nyi-Prasad-Sommerfield (BPS) states, we
find approximate BPS states which have non-zero masses that are almost
independent of the Yang-Mills coupling constant and which are a reflection of
the massless BPS states of the underlying N=1 super Yang-Mills theory. The
masses of these states at large Yang-Mills coupling are exactly at the
n-particle continuum thresholds. This leads to a relation between their masses
at zero and large Yang-Mills coupling.
