Billi et al. (2017) proposed a new interpretation for the origin and internal structure of thermogene travertine deposits. On the basis of evidence from two quarries located in southern Tuscany (Italy), they interpreted some travertine beds as calcite veins and argued that undulating travertine beds formed by syn-diagenetic (i.e. non tectonic) folding that was caused by laterally-confined volume expansion caused by incremental veining. They assumed that such a process causes changes to the rock properties, including porosity reduction, rock strengthening, and age rejuvenation. The interpretations by Billi et al. (2017) challenge and question the current understanding and interpretation of thermogene travertine deposits. This understanding, based on numerous studies since the 1980s, is that these deposits form from thermal water flowing downslope, and precipitating calcium carbonate. Here, we explain how the comparison with active depositional systems is essential for the understanding the origin of structures in older, inactive travertine deposits, such as those studied by Billi et al. (2017). We further argue that the three-dimensional setting of travertine deposits should be taken into account in order to discuss the possible development of secondary structures. Indeed travertine deposition on slopes typically leads to the formation of terraced morphologies with pools bordered by rounded rims and separated from each other by steep walls. The resulting three-dimensional structures can be misinterpreted as asymmetric folds in two-dimensional views (i.e., in saw-cut walls of quarry). In this paper we debate the interpretations offered by Billi et al. (2017) and their criteria to recognise syn-diagenetic, non-tectonic folds in travertine deposits, and explain why many of their ideas are questionable.