  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 $\depth(M)+\depth(N)=\depth(R)+\depth(M\otimes_{R}N)$ to hold. We
show that, under certain conditions, $M$ and $N$ satisfy the depth formula if
and only if $\Tor_{i}^{R}(M,N)$ vanishes for all $i\geq 1$. We also examine the
relationship between good depth of $M\otimes_RN$ and the vanishing of $\Ext$
modules, with various applications.
