  (ABRIDGED) Gaseous halos of nine nearby galaxies (with redshifts cz < 6000
km/s) were probed at large galactocentric radii using background quasars
observed with HST GHRS and STIS. The projected quasar-galaxy separations range
from 55 to 387 h^{-1}_{75} kpc. Ly\alpha absorption lines were successfully
detected in the spectra of five quasars, at impact parameters of up to 170
h^{-1}_{75} kpc from the center of the nearby galaxy, and in each case at
wavelengths consistent with the galaxy's redshift. Our observations include the
lowest redshift Ly\alpha lines detected to date. HI velocity fields were
obtained at the VLA for three of the galaxies in our sample (and in one case
was available from the literature), to derive their rotation curves. When
comparing the inner rotation curves of the galaxies with the velocity at large
radius provided by the Ly\alpha line it is apparent that it is very difficult
to explain the observed Ly\alpha velocity as due to gas in an extended rotating
disk. In most cases one would need to invoke large warps in the outer gas disks
and also thick gas disks in order to reconcile the observed velocities with the
predicted ones. In light of these results we conclude that in a typical galaxy
there is no longer detectable atomic gas corotating in an extended disk at
radii > 35 \alpha ^{-1}, where \alpha^{-1} is the stellar disk exponential
scale-length. The cosmic web is the most likely origin for the detected
Ly\alpha lines.
