Worn key

 is a miscellaneous item in Fallout: New Vegas. There are two (and likely more) worn keys that can be found. They are likely independent and can not open other locks than the one they were designed for, although this has not been confirmed.

Primm
A worn key can be found on a dead escaped convict on the 3rd floor of the Bison Steve Hotel. It unlocks the gun cabinet found near the dead convict. You can access the third floor by entering one of the rooms on the 2nd floor that has a collapsed ceiling, and walking up the ceiling like you would a ramp.
 * If you repair the elevator, it will still only take you to the 2nd floor and you must still find the collapsed room to gain access to the 3rd floor.

Nipton
 is sitting on top of several papers on the counter behind the reception desk that is on the right side, just as you walk inside the Nipton Town Hall. The key unlocks the door to the basement in the Nipton Town Hall. This door is adjacent to the stairwell leading up to the second floor. It is located on the left side, not far in from the entrance to the Nipton Town Hall.
 * Be careful of the Frag mine (Fallout: New Vegas) that is on the floor next to the counter.
 * Also be careful of the Legion mongrels that roam the hall in the lobby.