We examine atypical current fluctuations in totally asymmetric zero-range processes in one dimension with periodic boundary conditions. The zero-range processes is a stochastic lattice gas in which each lattice site can be occupied by, a-priori, an unbounded number of particles. Particles move to their neighbour at a rate which only depends on the occupation of the departure site. For large systems, by calculating the Jensen-Varadhan action functional, we are able to find the time dependent optimal profiles which realise currents below the typical value. Under certain conditions on the jump rates, we demonstrate that these systems can exhibit a dynamical phase transition, in which above a critical non-typically current the optimal macroscopic density profile is given by a traveling wave with a shock and anti-shock pair. While rare events below the critical current are realised by a condensate, whereby a non-zero fraction of all the particles accumulate on a single site in the thermodynamic limit. This gives rise to a non-convex rate function for the empirical current, which in turn leads to a breakdown of the equivalence between the conditioned dynamics and the s-ensemble cloning methods typically used in simulations to sample these rare events.