Many animals have socially transmitted behavioural traditions, but human culture appears unique in that it is cumulative, i.e. human cultural traits increase in diversity and complexity over time. It is often suggested that high-fidelity cultural transmission is necessary for cumulative culture to occur through refinement, a process known as 'ratcheting', but this hypothesis has never been formally evaluated. We discuss processes of information transmission and loss of traits from a cognitive viewpoint alongside other cultural processes of novel invention (generation of entirely new traits), modification (refinement of existing traits) and combination (bringing together two established traits to generate a new trait). We develop a simple cultural transmission model that does not assume major evolutionary changes (e.g. in brain architecture) and show that small changes in the fidelity with which information is passed between individuals can lead to cumulative culture. In comparison, modification and combination have a lesser influence on, and novel invention appears unimportant to, the ratcheting process. Our findings support the idea that high-fidelity transmission is the key driver of human cumulative culture, and that progress in cumulative culture depends more on trait combination than novel invention or trait modification.