Why CIWS would struggle with battleship salvos

Could modern anti-aircraft and close-in weapon systems shoot down a battleship shell?

They might engage individual shells, but the speed, trajectory and number in a full salvo would probably overwhelm their ammunition capacity.

The problem involves both interception and magazine depth. A 16-inch shell arrives fast and steeply, while a battleship may send nine or more rounds together. Even systems able to engage one projectile could run out of ready ammunition against the salvo.