We study resonant tunneling through a layered medium with a passive negative index medium (NIM) slab as a constituent layer. Using a causal
model for susceptibilities with the parameters of a recently reported metamaterial [G. Dolling, C. Enkrich, M. Wegener, C.M. Soukoulis, S. Linden, Opt. Lett. 31, 1800 (2006)] we show that resonant tunnelling and the associated delay are mostly suppressed. This is in sharp contrast with the naive approach of retaining phase velocity dispersion with arbitrary low losses, predicting sharp resonances with large associated delays. This is shown to be a nontrivial issue because of the necessity of losses for NIM behaviour, while their presence spoils the quality factor of the resonant devices.