In the assessment of volcanic risk, it is often assumed that magma ascending at a slow rate will erupt effusively, whereas magma ascending at fast rate will lead to an explosive eruption. Mechanistically viewed, this assessment is supported by the notion that the viscoelastic nature of magma (i.e., the ability of magma to relax at an applied strain rate), linked via the gradient of flow pressure (related to discharge rate), controls the eruption style. In such an analysis, the physical interactions between the magma and the conduit wall are commonly, to a first order, neglected. Yet, during ascent, magma must force its way through the volcanic edifice/structure, whose presence and form may greatly affect the stress field through which the magma is trying to ascend. Here, we demonstrate that fracturing of the conduit wall via flow pressure releases an elastic shock resulting in fracturing of the viscous magma itself. We find that magma fragmentation occurred at strain rates seven orders of magnitude slower than theoretically anticipated from the applied axial strain rate. Our conclusion, that the discharge rate cannot provide a reliable indication of ascending magma rheology without knowledge of conduit wall stability, has important ramifications for volcanic hazard assessment. New numerical simulations are now needed in order to integrate magma/conduit interaction into eruption models.