  We expound in detail the degeneracy between the octant of $\theta_{23}$ and
flavor-changing neutral-current non-standard interactions (NSI's) in neutrino
propagation, considering the Deep Underground Neutrino Experiment (DUNE) as a
case study. In the presence of such NSI parameters involving the $e-\mu$
($\varepsilon_{e\mu}$) and $e-\tau$ ($\varepsilon_{e\tau}$) flavors, the
$\nu_\mu \to \nu_e$ and $\bar\nu_\mu \to \bar\nu_e$ appearance probabilities in
long-baseline experiments acquire an additional interference term, which
depends on one new dynamical CP-phase $\phi_{e\mu/e\tau}$. This term sums up
with the well-known interference term related to the standard CP-phase $\delta$
creating a source of confusion in the determination of the octant of
$\theta_{23}$. We show that for values of the NSI coupling (taken one
at-a-time) as small as $few\,\%$ (relative to the Fermi coupling constant
$G_{\mathrm F}$), and for unfavorable combinations of the two CP-phases
$\delta$ and $\phi_{e\mu/e\tau}$, the discovery potential of the octant of
$\theta_{23}$ gets completely lost.
