Let \(G\) be a reduced abelian \(p\)-group of length \(\lambda\), let \({\mathcal E}(G)\) denote the endomorphism ring of \(G\), and let \({\mathbf J}(G)\) be the Jacobson radical of \({\mathcal E}(G)\). For any ordinal \(\kappa \leq \lambda\), define \({\mathbf H}_ \kappa (G)\) to be the set of all \(\eta \in {\mathcal E} (G)\) with the property that, for all ordinals \(\mu < \kappa\), \(p^ \mu G[p] \eta \leq p^{\mu+1} G\). \textit{R. S. Pierce} has shown that \({\mathbf H}_{\omega} (G)\) is always an upper bound for \({\mathbf J}(G)\) with equality possible but not necessary [Topics in Abelian Groups, Scott, Foresman, and Co., Glenview, IL, 215- 310 (1963)]. This led to a series of papers characterizing \({\mathbf J} (G)\) in terms of its action on \(G[p]\) for \(G\) belonging to various classes of \(p\)-groups. In all cases considered, \({\mathbf J} (G) \subseteq {\mathbf H}_ \lambda (G)\). The purpose of the present article is to show that \(\mathbf J(G)\) need not be contained in \({\mathbf H}_{\lambda} (G)\). A sufficient condition is given for when \({\mathbf J} (G) \subseteq {\mathbf H}_ \lambda (G)\).