The observation that the b --> sgamma decay rate is close to the standard model value implies a large mass for the charged Higgs boson in the minimum supersymmetric standard model (MSSM), which nearly closes the t --> bH+ decay channel. For m(t) = 150 GeV, the parameter region m(A) less-than-or-similar-to 130 GeV is excluded; this largely preempts searches at the CERN e+e- collider LEP II and also partially excludes a region that would be inaccessible to MSSM Higgs boson searches at LEP II and at the Superconducting Super Collider and the CERN Large Hadron Collider.