It's not possible be truly double-jointed. People who appear to be double-jointed actually have a form of hypermobility, which means that they are able to extend their joints to the full range of motion without experiencing pain.
It is not possible to develop hypermobility through any type of training as it is believed to be a genetic trait that may also have to do with the shapes of the bones and joints themselves. The condition can be present in several different joints. Hypermobility is not necessarily a good condition to have and can cause pain and an increased risk of arthritis.