Humans' distinctive role in the world can largely be attributed to our capacity for iterated learning, a process by which knowledge is expanded and refined over generations. A range of theories seek to explain why humans are so adept at iterated learning, many positing substantial evolutionary discontinuities in communication or cognition. Is it necessary to posit large differences in abilities between humans and other species, or could small differences in communication ability produce large differences in what a species can learn over generations? We investigate this question through a formal model based on information theory. We manipulate how much information individual learners can send each other and observe the effect on iterated learning performance. Incremental changes to the channel rate can lead to dramatic, non-linear changes to the eventual performance of the population. We complement this model with a theoretical result that describes how individual lossy communications constrain the global performance of iterated learning. Our results demonstrate that incremental, quantitative changes to communication abilities could be sufficient to explain large differences in what can be learned over many generations.