Matěj Jan Morávek
In rulebook it specifically says that (p. 18):
Each investigator can only attempt the same puzzle once per turn (but multiple investigators may attempt the same puzzle on the same turn).
You can try it only once. However, as I understand, the lock puzzle on the room is attempted using movement points, not actions. So when you or anyone else tries to continue, you (he/she) will spend one movement point, not an action.
The unsolved puzzle stays in its state until it is solved or some effect makes it "reset" (Keeper can do that with certain mythos cards). So when you (next turn) or another investigator tries the puzzle again, you continue where your predecessor left.