The Sponge function is known to achieve \( 2^{c/2} \) security, where \( c \) is its capacity. This bound was carried over to its keyed variants, such as SpongeWrap, to achieve a \( \min( 2^{c/2},2^{\kappa}) \) security bound, with \( \kappa \) the key length. Similarly, many CAESAR competition submissions were designed to comply with the classical \( 2^{c/2} \) security bound. We show that Sponge-based constructions for authenticated encryption can achieve the significantly higher bound of \( \min( 2^{b/2},2^{c},2^{\kappa}) \), with \( b>c \) the permutation size, by proving that the CAESAR submission NORX achieves this bound. The proof relies on rigorous computation of multi-collision probabilities, which may be of independent interest. We additionally derive a generic attack based on multi-collisions that matches the bound. We show how to apply the proof to five other Sponge-based CAESAR submissions: Ascon, CBEAM/STRIBOB, ICEPOLE, Keyak, and two out of the three PRIMATEs. A direct application of the result shows that the parameter choices of some of these submissions are overly conservative. Simple tweaks render the schemes considerably more efficient without sacrificing security. We finally consider the remaining one of the three PRIMATEs, APE, and derive a blockwise adaptive attack in the nonce-respecting setting with complexity \( 2^{c/2} \), therewith demonstrating that the techniques cannot be applied to APE.