The main purpose of this paper is to englobe some new and known types of Hermitian block-matrices $M=\begin{pmatrix} A & X\\ {X^*} & B\end{pmatrix}$ satisfying or not the inequality $\|M\|\le \|A+B\|$ for all symmetric norms. For positive definite block-matrices another inequality is established and it is shown that it can be sharper (for some symmetric norms) than the following holding inequality $\|M\|\le \|A\|+\|B\|$.