  In QFT, the null energy condition (NEC) for a classical field configuration
is usually associated with that configuration's stability against small
perturbations, and with the sub-luminality of these. Here, we exhibit an
effective field theory that allows for stable NEC-violating solutions with
exactly luminal excitations only. The model is the recently introduced
`galileon', or more precisely its conformally invariant version. We show that
the theory's low-energy S-matrix obeys standard positivity as implied by
dispersion relations. However we also show that if the relevant NEC-violating
solution is inside the effective theory, then other (generic) solutions allow
for superluminal signal propagation. While the usual association between
sub-luminality and positivity is not obeyed by our example, that between NEC
and sub-luminality is, albeit in a less direct way than usual.
