In this work we present bounds on CPT invariance using global neutrino oscillation data and then analyze how much DUNE could improve these bounds. We also find that if CPT is not conserved in nature, we are actually obtaining fake solutions for the neutrino oscillation parameters. To give a concrete example we show that, if the recent results from the T2K collaboration turn out to be the true values of neutrino and antineutrino oscillations, DUNE would measure CPT violation at more than 3$\sigma$. We also show that these results could not be obtained by neutrino non-standard interactions.