Every casino does have a rule which covers this very problem. Usually, what they'll do is to leave all the live hands ON the table, & take the cards from the discard tray, shuffle them, burn a card, and continue play from that point. The reason that the dealer was worried is because he would've probably gotten some flak from his boss, for starting a round without having enough cards to finish. It matters not whether or not it was his fault. Casino dealers are quite often treated like s**t by their superiors.