  We investigate the possible reasons for the discrepancy between the
theoretical two-photon ionization cross section, $\sim 10^{-56}$ $\text{cm}^4
\text{s}$, of Ne$^{8+}$ obtained within the perturbative nonrelativistic
framework for monochromatic light [J. Phys. B {\bf 34}, 4857 (2001)] and the
experimental value, $7 \times 10^{-54}$ $\text{cm}^4 \text{s}$, reported in
[Phys. Rev. Lett. {\bf 106}, 083002 (2011)] at a photon energy of 1110 eV. To
this end, we consider Ne$^{8+}$ exposed to deterministic and chaotic ensembles
of intense x-ray pulses. The time-dependent configuration-interaction singles
(TDCIS) method is used to quantitatively describe nonlinear ionization of
Ne$^{8+}$ induced by coherent intense ultrashort x-ray laser pulses. The impact
of the bandwidth of a chaotic ensemble of x-ray pulses on the effective
two-photon ionization cross section is studied within the lowest nonvanishing
order of perturbation theory. We find that, at a bandwidth of 11 eV, the
effective two-photon ionization cross section of Ne$^{8+}$ at a photon energy
of 1110 eV amounts to $5 \times 10^{-57}$ and $1.6 \times 10^{-55}$
$\text{cm}^4 \text{s}$ for a deterministic ensemble and a chaotic ensemble,
respectively. We show that the enhancement obtained for a chaotic ensemble of
pulses originates from the presence of the one-photon 1$s^2$--1$s4p$ resonance
located at 1127 eV. Using the TDCIS approach, we also show that, for currently
available radiation intensities, two-photon ionization of a 1$s$ electron in
neutral neon remains less probable than one-photon ionization of a valence
electron.
