  In this paper we analyse the possibility of having homogeneous isotropic
cosmological models with observers reaching $t=\infty$ in finite proper time.
It is shown that just observationally-suggested dark energy models with
$w\in(-5/3,-1)$ show this feature and that they are endowed with an exotic
curvature singularity. Furthermore, it is shown that non-accelerated observers
in these models may experience a duration of the universe as short as desired
by increasing their linear momentum. A subdivision of phantom models in two
families according to this behavior is suggested.
