  We compute an array of Standard Model Higgs boson ($\hsm$) signals and
backgrounds for a possible upgrade of the Tevatron to $E_{\rm cm}=4\tev$.
Taking $\mt\geq 140\gev$, and assuming a total accumulated luminosity of
$L=30\fbi$, we find that a Standard Model Higgs boson with $\mhsm\lsim 110\gev$
could almost certainly be detected using the $\wpm\hsm\rta l\nu b\anti b$ mode.
A Higgs boson with mass between $\sim 120\gev$ and $\sim 140\gev$ or above
$\sim 230-250\gev$ almost certainly would not be seen. A Higgs boson with
$\mhsm\sim 150\gev$ or $200\lsim\mhsm\lsim 230-250\gev$ has a decent chance of
being detected in the $ZZ\rta 4l$ mode. There would also be some possibility of
discovering the $\hsm$ in the $WW\rta l\nu jj$ mode for $150\lsim\mhsm\lsim
200\gev$. Finally, hints of an event excess in the $WW\rta ll \nu\nu$ mode due
to the $\hsm$ might emerge for $140\lsim\mhsm\lsim 180\gev$. Given the
difficult nature of the Higgs boson signals for $\mhsm$ values beyond the reach
of LEP-200, and the discontinuous $\mhsm$ range that could potentially be
probed, justification of an upgrade of the Tevatron to $4\tev$ on the basis of
its potential for Standard Model Higgs boson discovery would seem
inappropriate.
