Volcanic uplift, caused by the accumulation of magma in subsurface reservoirs, is a common precursor to eruptions. But, for some volcanoes, uplift of metres or more has not yet led to an eruption. Here we present displacement maps of volcanoes in the Galápagos Islands, constructed using satellite radar interferometry, that might help explain this dichotomy. We show that all but one of the seven volcanoes on the islands of Isabela and Fernandina deformed during 1992-99. Cerro Azul and Fernandina erupted during the observation period and show evidence of inflation, co-eruptive deflation and shallow dyke intrusion. In contrast, the largest volcano, Sierra Negra, has not erupted, yet exhibits spatially and temporally variable deformation, with a maximum uplift of 2.7 m between 1992 and 1999, which can be modelled by a shallow inflating sill. Inflation during 1997-98, however, was accompanied by 'trapdoor' faulting on a steeply dipping fracture system within the caldera. Repeated trapdoor faulting over geological time has formed an arcuate intra-caldera ridge within Sierra Negra and may have acted to relax stresses above the magma chamber, inhibiting summit eruptions. Similar processes may help explain large uplift unaccompanied by eruptive activity at other volcanoes.