In this paper we prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring (A, m), and if H i m (M ) = 0 except possibly for i ∈ {0, r, d} with some 0 ≤ r ≤ d, then there exists an integer such that every parameter ideal for M contained in m has the same index of reducibility. This theorem generalizes earlier work of the second author and is closely related to recent work of Goto-Suzuki and Goto-Sakurai; Goto-Sakurai have supplied an answer of yes in case M is Buchsbaum.