Let $R$ be a Noetherian ring and $M,N,U$ be $R$-modules. We have a short exact sequence
$$0 \longrightarrow U \longrightarrow M \longrightarrow N \longrightarrow 0.$$
We know that $\operatorname{depth}(U) \geq \min \{\operatorname{depth}(M),\operatorname{depth}(N)+1\}$.
Suppose $\operatorname{depth}(M)=\operatorname{depth}(N)$.
Can I say that $\operatorname{depth}(U)=\operatorname{depth}(N)$ ?
No. We always have $"\ge"$, but it happens to have $">"$ as the following example shows.
Set $R=K[X]_{(X)}$ and consider the short exact sequence of $R$-modules $$0\to R\to K\oplus R\to K\to 0.$$