We apply a chemical evolution model to investigate the sources and evolution of dust in a sample of 26 high-redshift (z > 1) submillimetre galaxies (SMGs) from the literature, with complete photometry from ultraviolet to the submillimetre. We show that dust produced only by low–intermediate mass stars falls a factor 240 short of the observed dust masses of SMGs, the well-known ‘dust-budget crisis’. Adding an extra source of dust from supernovae can account for the dust mass in 19 per cent of the SMG sample. Even after accounting for dust produced by supernovae the remaining deficit in the dust mass budget provides support for higher supernova yields, substantial grain growth in the interstellar medium or a top-heavy IMF. Including efficient destruction of dust by supernova shocks increases the tension between our model and observed SMG dust masses. The models which best reproduce the physical properties of SMGs have a rapid build-up of dust from both stellar and interstellar sources and minimal dust destruction. Alternatively, invoking a top-heavy IMF or significant changes in the dust grain properties can solve the dust budget crisis only if dust is produced by both low mass stars and supernovae and is not efficiently destroyed by supernova shocks.