  Let $I$ denote an ideal of a Noetherian ring $R$. The purpose of this article
is to introduce the concepts of quintasymptotic sequences over $I$ and
quintasymptotic cograde of $I$, and it is shown that they play a role analogous
to quintessential sequences over $I$ and quintessential cograde of $I$, given
in \cite{Ra1}. Also, we show that, if $R$ is local, then the quintasymptotic
cograde of $I$ is unambiguously defined and behaves well when passing to
certain related local rings. Finally, we use this cograde to characterize of
two classes of local rings.
