Lower bounds for 3SUM with a free cache
Well, sure, it's possible.  But since we basically don't know how to prove that anything is hard, it doesn't look easy.

In the mid 90s I proved an $\Omega(n^2)$ lower bound for 3SUM in a variant of the linear decision tree model; my proof was later simplified by Ailon and Chazelle.  This model of computation has nothing corresponding to "memory"; the running time of an algorithm is defined simply as the worst-case number of (generalized) comparisons.  So adding a cache won't break these lower bounds; to beat $O(n^2)$ time, you have to do some non-obvious computation with the data in the cache.