Let $M$ and $N$ be finitely generated modules over a codimension $c$ complete intersection $R$. We show for $c=2$ or $3$, and under certain mild assumptions (which escalate with higher codimension), that if the tensor product $M \otimes_R N$ satisfies Serre's condition $(S_{c+1})$, then $\Tor^R_n(M,N)=$ for all $n>0$.