  The exchange of a pair of neutrinos with Standard Model weak interactions
generates a long-range force between fermions. The associated potential is
extremely feeble, $\propto G_F^2/r^5$ for massless neutrinos, whichrenders it
far from observable even in the most sensitive experiments testing fifth
forces. The presence of a neutrino background has been argued to induce a
correction to the neutrino propagator that enhances the potential by orders of
magnitude. In this brief note, we point out that such modified propagators are
invalid if the background neutrino wavepackets have a finite width. By
reevaluating the 2--$\nu$ exchange potential in the presence of a neutrino
background including finite width effects, we find that the background-induced
enhancement is reduced by several orders of magnitude. Unfortunately, this
pushes the resulting 2--$\nu$ exchange potential away from present and
near-future sensitivity of tests of new long-range forces.
