Abstract Explosive volcanic eruptions occur when coherent bubbly magma breaks apart in a process called ‘fragmentation.’ Accurate conceptual and numerical models of fragmentation are a pre‐requisite for prediction of eruption explosivity. However, existing models, which neglect bubble‐scale magma physics, agree poorly with new experimental evidence that we present. Here, we derive and validate a mathematical model for magma fragmentation, based on viscoelastic failure of the liquid around rapidly growing bubbles in magma. The model, which explicitly captures processes at the bubble scale, shows excellent agreement with experiments, accurately predicting the conditions under which magma does and does not fragment. We apply the new numerical model to predict the critical rate of magma decompression during ascent required for explosive eruption across different magma types. The predicted rates are consistent with chemical and textural proxies for the decompression rate, providing further validation of the model under natural eruption conditions.