Thanks for your help @HailongDao, it was under my eyes... With this settings the answer is NO! as Example 1.4 shows. Let be $R:=k[x_1,x_2,x_3,x_4$ and $I=(x_1x_2, x_2x_3, x_3x_4)\subseteq R.$ $I$ is an aCM ideal of height 2 so $depth R/I=2,$ but one can check that $depth \frac{R}{I+(x_2+x_3)}=0.$

@HailongDao Thanks for the question. The main motivation, currently, came from Proposition 1.2 in arxiv.org/pdf/0812.2080.pdf This implies that if $depth \frac{R}{I+(x_1, \ldots, x_t)}=0$ then $depth \frac{R}{I}\le t.$ On the other hand, if $\{\ell_1,\ldots, \ell_t \}$ is an $R/I$-sequence the statement is trivially true.