At 1105 Pacific time, a Piper PA-44-180 struck the ground during an instrument departure from Livermore Municipal Airport. The flight instructor and instrument-rated student were not injured. The flight departed on a standard instrument departure and were cleared to 4,000 feet. The instructor said they then responded to a call to maintain 2,000 feet, and continued with the SID. While intersecting the OAK 060 radial, the instructor saw terrain through the clouds and took evasive action. They struck something, but the airplane was still controllable so they proceeded to their destination of Sacramento Executive Airport. A review of the tapes of radio calls found that the pilots apparently thought the 2,000-foot clearance was meant for them, but it was directed at another aircraft in another sector being worked by the same controller.