Can I ever run out of lockpicks?

Solution 1:

Lockpicks always unlock optional/bonus areas.

You'll notice that certain locks are marked "hairpin" and this means that Elizabeth can open them without needing any lockpicks. These locks are almost always story related, and are just there to keep you from skipping the combat sections. Elizabeth won't pick a lock while you're in a fight.

As far as these optional areas go, I've only ever seen one safe that I had trouble opening with my current lockpick count, and it was very soon after getting Elizabeth. Every other instance of pickable locks I've found there have been plenty of lockpicks to open. In fact, eventually I hit the cap of 30 later in the game.