Almost always.
You can get into The Cave anytime with advance notice. The best way to secure entry beyond posted hours is to call The Cave during regular business hours and make an appointment. 818-242-5192 -Wed - Sun, 2-7 pm.
In addition to this, there is the super-secret bat phone option, and that phone number is 818-359-5182. Because I live in the building, if I'm around you're as good as in. There is always the possibility that I may be out or otherwise unavailable, so your best bet is advance notice. But give it a shot, it's been pretty successful so far.
Never an early-by-nature riser, after noon increases your chances.
(If you call at 8 a.m. thinking you might get in at nine, probably not, but if you call during business hours the day before and want to get in at nine, no problem.)
This is a complimentary service.