Question:
what upper bounds are known on the number of non-metric entries of finite dimensional square matrices $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ with strictly positive off-diagonal elements $a_{ij}$?

In this context $a_{ij}$ is defined to metric iff $\quad a_{ij}\leqq a_{ik}+a_{kj}\,\forall k\notin\lbrace i, j\rbrace\quad $ and non-metric otherwise.