Since the late 1990s, rising sea levels around the Torres Islands (north Vanuatu, southwest Pacific) have caused strong local and international concern. In 2002-2004, a village was displaced due to increasing sea incursions, and in 2005 a United Nations Environment Programme press release referred to the displaced village as perhaps the world's first climate change "refugees." We show here that vertical motions of the Torres Islands themselves dominate the apparent sea-level rise observed on the islands. From 1997 to 2009, the absolute sea level rose by 150 + /-20 mm. But GPS data reveal that the islands subsided by 117 + /-30 mm over the same time period, almost doubling the apparent gradual sea-level rise. Moreover, large earthquakes that occurred just before and after this period caused several hundreds of mm of sudden vertical motion, generating larger apparent sea-level changes than those observed during the entire intervening period. Our results show that vertical ground motions must be accounted for when evaluating sea-level change hazards in active tectonic regions. These data are needed to help communities and governments understand environmental changes and make the best decisions for their future.