  From the LEP precision data and the measurement of the W-boson mass, upon
excluding the observables Rb, Rc in a combined fit of the top-quark mass, Mt,
and the Higgs-boson mass, MH, within the Standard Model, we find the weak
1sigma bound of MH<900GeV. Stronger upper bounds on MH, sometimes presented in
the literature, rely heavily on the inclusion of Rb in the data sample. Upon
including Rb, the quality of the fit drastically decreases, and by carefully
analyzing the dependence of the fit results on the set of experimental input
data we conclude that these stronger bounds are not reliable. Moreover, the
stronger bounds on MH are lost if the deviation between theory and experiment
in Rb is ascribed to contributions of new physics. Replacing swbar^2(LEP) by
the combined value swbar^2(LEP+SLD) in the data sample leads to a bound of
MH<430GeV at the 1sigma level. The value of swbar^2(SLD) taken alone, however,
gives rise to fit results for MH which are in conflict with MH<65.2GeV from
direct searches.
