  We determine the minimal lower bound $n$, with $n \geq 1$, where the $n$-th
power of the radical of the module category of a representation-finite cluster
tilted algebra vanishes. We give such a bound in terms of the number of
vertices of the underline quiver. Consequently, we get the nilpotency index of
the radical of the module category for representation-finite self-injective
cluster tilted algebras. We also study the non-zero composition of $m$, $m \ge
2$, irreducible morphisms between indecomposable modules in
representation-finite cluster tilted algebras lying in the $(m+1)$-th power of
the radical of their module category.
