Fragmentation of vesicular magma by rapid decompression is one of the most likely triggers for explosive eruptions. In this phenomenon the decompression rate and the viscoelastic nature of magma are considered to be key factors. In order to obtain a clear idea on the effects of these two factors, controlled fragmentation experiments have been conducted. These experiments have three advantages. First, the specimen is made of a viscoelastic material with controlled porosity and geometry. Second, the fragmentation process is directly monitored. Finally, both the magnitude and rate of decompression are controlled. Brittle fragmentation and ductile expansion were both observed in the same porous material at different timescales. The various mechanical responses of the specimen (elastic, flow, and fragmentation) were correlated with the pressure profile measured at the base of the specimen. Fragmentation was noted to occur when the decompression exceeded a critical value within a critical time. Two relevant timescales are discussed in terms of physical mechanisms of relaxation. The first is the measured glass transition time. The second is the estimated timescale for the onset of viscous bubble expansion. The observed phenomena bear several similarities with natural magma fragmentation. It is thus considered that the present results are a useful step toward constructing a model for magma fragmentation.