I wrote “rarish” where “extremely rare” might have been more accurate, but it is wrong to claim it is impossible. If you are dealt a pair of deuces in a single-deck game, it is possible to consume 20 cards just for that one initial hand by repeated splitting and hitting (*). (Even more than 20 cards could be consumed in a double-deck game, since you might split more than thrice.) Assuming dealer will deal to three players with as few as 32 cards available, and the player hands include the Deuces just described and a Pair of Aces resplit to four hands, the cards will run out if either the 3rd player or dealer needs a hit.
IIRC, I only saw the need for mid-hand shuffle once or twice in all the Blackjack I played, but these were with single-deck, while I played mostly against 4 decks or more where the rarity is even more extreme. 
Admittedly, the few times I saw it happen may have arose because of dealer oversight: overlooking that fewer than 30 cards remain (or whatever the house’s threshold is).
(* - Actually, up to 24 cards can be consumed if, perversely to demonstrate the possibility of run-out, player hits hard 20! 
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