We discuss recent findings suggesting that non-human animals lack memory for stimulus sequences, and therefore do not represent the order of stimuli faithfully. These observations have far-reaching consequences for animal cognition, neuroscience, and studies of the evolution of language and culture. This is because, if non-human animals do not remember or process information about order faithfully, then it is unlikely that non-human animals perform mental simulations, construct mental world models, have episodic memory, or transmit culture faithfully. If this suggested sequence bottleneck proves to be a prevalent characteristic of animal memory systems, as suggested by recent work, it would require a re-examination of some influential concepts and ideas.