  We investigate the bounds on the sum of neutrino masses in a
cosmic-acceleration scenario where the equation of state $w(z)$ of dark energy
(DE) is constructed in a model-independent way, using a basis of principal
components (PCs) that are allowed to cross the phantom barrier $w(z)=-1$. We
find that the additional freedom provided to $w(z)$ means the DE can undo
changes in the background expansion induced by massive neutrinos at low
redshifts. This has two significant consequences: (1) it leads to a substantial
increase in the upper bound for the sum of the neutrino masses ($M_{\nu} < 0.33
- 0.55$ eV at the 95\% C.L. depending on the data sets and number of PCs
included) compared to studies that choose a specific parametrization for
$w(z)$; and (2) it causes $\sim1\sigma$ deviations from $\Lambda$CDM in the
luminosity distance and the Hubble expansion rate at higher redshifts ($z
\gtrsim 2$), where the contribution of DE is subdominant and there is little
constraining data. The second point consequently means that there are also
observable deviations in the shear power spectrum and in the matter power
spectrum at low redshift, since the clustering of matter throughout cosmic time
depends on the expansion rate. This provides a compelling case to pursue
high-$z$ BAO and SN measurements as a way of disentangling the effects of
neutrinos and dark energy. Finally, we find that the additional freedom given
to the dark energy component has the effect of lowering $S_8$ with respect to
$\Lambda$CDM.
