Applying tests for jumps to financial data sets can lead to an important number of spurious detections. Bursts of volatility are often incorrectly identified as jumps when the sampling is too sparse. At a higher frequency, methods robust to microstructure noise are required. We argue that whatever the jump detection test and the sampling frequency, a highly elevant number of spurious detections remain because of multiple testing issues. We propose a formal treatment based on an explicit thresholding on available test statistics. We prove that our method eliminates asymptotically all remaining spurious detections. In Dow Jones stocks between 2006 and 2008, spurious detections can represent up to 90% of the jumps detected initially. For the stocks considered, jumps are rare events, they do not cluster in time, and no cojump affects all stocks simultaneously, suggesting jump risk is diversifiable. We relate the remaining jumps to macroeconomic news, prescheduled company-specific announcements, and stories from news agencies which include a variety of unscheduled and uncategorized events. The vast majority of news do not cause jumps but may generate a market reaction of the form of bursts of volatility.