Only if the cost of the services is above the deductible. Let's say you have a $2500 deductible, and go in for a CAT scan that, coincidentally, works out to a cost of $2450. That all comes out of your pocket, if you're doing this on Jan 1 and haven't used any of your deductible yet, with the insurance company paying nothing. If your pockets are empty, the hospital doesn't get paid without hounding you, working out a payment plan, or going to collections.

As a sidebar, it's awesome when you have a situation as described, and then never go to the doctor again for the rest of the year, and so get no cost reductions on anything, despite paying out-of-pocket for the services you did use, plus whatever your premiums are.