When attempting to pick a lock, your lockpick will break if you try to pick unsuccessfully three times. To avoid breaking the pick, exit out after two attempts and then re-enter the lockpicking screen. There are hundreds of doors to pick in the game. Putting points into the Lockpicking skill will obviously help for harder locks. For a safe that can be picked and locked indefinitely, head to Farragut West Metro Station. You can pick the safe found here and then lock it via the nearby terminal, then pick it again and repeat until you get the achievement.
Note for PC Users: If you don't feel like finding the safe listed above, use the same console command process as for "Data Miner" but instead click on the door/locker/safe you want to lock and subsequently pick. Repeat until unlocked.