We show that for a finitely generated module $M$ over a complete intersection $R$, the vanishing of $\Ext^i_R(M,M)$ for a certain number of consecutive values of $i$ starting at $n$ forces the projective dimension of $M$ to be at most $n-1$. In particular, $\Ext^2_R(M,M)=0$ if and only if the projective dimension of $M$ over $R$ is at most one. We also verify a conjecture of Auslander and Reiten for modules over commutative rings with certain typical behaviors, which augments the recent literature.