  Let $G$ be a graph on the vertex set $[n]$ and $J_G$ be the associated
binomial edge ideal in the polynomial ring
$S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$. In this paper we investigate
about the depth of binomial edge ideals. More precisely, we first establish a
combinatorial lower bound for the depth of $S/J_G$ based on some graphical
invariants of $G$. Next, we combinatorially characterize all binomial edge
ideals $J_G$ with $\mathrm{depth}\hspace{1.2mm}S/J_G=5$. To achieve this goal,
we associate a new poset $\mathcal{M}_G$ to the binomial edge ideal of $G$, and
then elaborate some topological properties of certain subposets of
$\mathcal{M}_G$ in order to compute some local cohomology modules of $S/J_G$.
