In this work, we consider the effect of non-standard interactions (NSIs) on the propagation of neutrinos through matter and how they in turn affect the sensitivity of the Deep Underground Neutrino Experiment (DUNE) in determining the mass hierarchy. We emphasize on the special case - when the diagonal NSI parameter $\epsilon_{ee} = -1$ at which the standard matter effect gets nullified. We show that, if in addition, there is maximal CP violation then this gives rise to an exact intrinsic hierarchy degeneracy, in the appearance channel, irrespective of the baseline and energy. We learn that the introduction of the off-diagonal NSI parameter, $\epsilon_{e \tau}$, shifts the position of the intrinsic degeneracy to a different $\epsilon_{ee}$. Moreover the unknown magnitude and phases of the off-diagonal NSI parameters can give additional degeneracies. Considering current model independent bounds on NSI parameters we observe that the hierarchy sensitivity of DUNE gets seriously compromised if NSI exists in nature. A signal of neutrino mass hierarchy at DUNE will therefore be able to rule out certain ranges of the NSI parameters. Additionally, we discuss the implications of $\epsilon_{ee} = -1$ (in the Earth) on the MSW effect in the Sun.