Necessary conditions for the depth formula over Cohen-Macaulay local rings

Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula $\text{depth}(M)+\text{depth}(N)=\text{depth}(R)+\text{depth}(M{\otimes }_{R}N)$ to hold. We show that, under certain conditions, $M$ and $N$ satisfy the depth formula if and only if ${\text{Tor}}_i^R(M,N)$ vanishes for all $i\ge 1$. We also examine the relationship between good depth of $M{\otimes }_{R}N$ and the vanishing of $\text{Ext}$ modules, with various applications.