Sublinear time algorithms for approximating maximum matching size have long been studied. Much of the progress over the last two decades on this problem has been on the algorithmic side. For instance, an algorithm of Behnezhad [FOCS'21] obtains a 1/2-approximation in $\tilde{O}(n)$ time for $n$-vertex graphs. A more recent algorithm by Behnezhad, Roghani, Rubinstein, and Saberi [SODA'23] obtains a slightly-better-than-1/2 approximation in $O(n^{1+ε})$ time. On the lower bound side, Parnas and Ron [TCS'07] showed 15 years ago that obtaining any constant approximation of maximum matching size requires $Ω(n)$ time. Proving any super-linear in $n$ lower bound, even for $(1-ε)$-approximations, has remained elusive since then. In this paper, we prove the first super-linear in $n$ lower bound for this problem. We show that at least $n^{1.2 - o(1)}$ queries in the adjacency list model are needed for obtaining a $(\frac{2}{3} + Ω(1))$-approximation of maximum matching size. This holds even if the graph is bipartite and is promised to have a matching of size $Θ(n)$. Our lower bound argument builds on techniques such as correlation decay that to our knowledge have not been used before in proving sublinear time lower bounds. We complement our lower bound by presenting two algorithms that run in strongly sublinear time of $n^{2-Ω(1)}$. The first algorithm achieves a $(\frac{2}{3}-ε)$-approximation; this significantly improves prior close-to-1/2 approximations. Our second algorithm obtains an even better approximation factor of $(\frac{2}{3}+Ω(1))$ for bipartite graphs. This breaks the prevalent $2/3$-approximation barrier and importantly shows that our $n^{1.2-o(1)}$ time lower bound for $(\frac{2}{3}+Ω(1))$-approximations cannot be improved all the way to $n^{2-o(1)}$.