Deep neural networks (DNNs) excel at visual recognition tasks and are increasingly used as a modeling framework for neural computations in the primate brain. Just like individual brains, each DNN has a unique connectivity and representational profile. Here, we investigate individual differences among DNN instances that arise from varying only the random initialization of the network weights. Using tools typically employed in systems neuroscience, we show that this minimal change in initial conditions prior to training leads to substantial differences in intermediate and higher-level network representations despite similar network-level classification performance. We locate the origins of the effects in an under-constrained alignment of category exemplars, rather than misaligned category centroids. These results call into question the common practice of using single networks to derive insights into neural information processing and rather suggest that computational neuroscientists working with DNNs may need to base their inferences on groups of multiple network instances.