Breakdowns occurring in RF accelerating structures will limit the ultimate performance of future linear colliders such as CLIC. Because of the similarity of many aspects of DC and RF breakdown, a DC breakdown study is underway at CERN to better understand the vacuum breakdown mechanism in a simple setup. Measurements of the field enhancement factor beta show that the local breakdown field is constant and depends only on the electrode material. With copper electrodes, the local breakdown field is around 10.8 GV/m, independent of the gap distance. In breakdown rate experiments, clusters of consecutive breakdowns alternate with quiet periods. The occurrence and lengths of these clusters and quiet periods depend on the evolution of beta. The application of a high field can even modify the electrode surface in the absence of breakdown. Measurements of time delays to breakdown and of gas release during breakdown emphasize also the crucial role of field emission in the breakdown triggering.