Combinatorial communication, in which two signals are used together to achieve an effect that is different to the sum of the effects of the component parts, is apparently rare in nature: it is ubiquitous in human language, appears to exist in a simple form in some non-human primates, but has not been demonstrated in other species. This observed distribution has led to the pair of related suggestions, that (i) these differences in the complexity of observed communication systems reflect cognitive differences between species; and (ii) that the combinations we see in non-human primates may be evolutionary pre-cursors of human language. Here we replicate the landmark experiments on combinatorial communication in non-human primates, but in an entirely different species, unrelated to humans, and with no higher cognition: the bacterium Pseudomonas aeruginosa. Using the same general methods as the primate studies, we find the same general pattern of results: the effect of the combined signal differs from the composite effect of the two individual signals. This suggests that advanced cognitive abilities and large brains do not necessarily explain why some species have combinatorial communication systems and others do not. We thus argue that it is premature to conclude that the systems observed in non-human primates are evolutionarily related to language. Our results illustrate the value of an extremely broad approach to comparative research.