Let $\lambda \leq \kappa$ be infinite cardinals and let $\Omega$ be a set of cardinality $\kappa$. The bounded permutation group B$_\lambda(\Omega)$, or simply B$_\lambda$, is the group consisting of all permutations of $\Omega$ which move fewer than $\lambda$ points in $\Omega$. We say that a permutation group G acting on $\Omega$ is a supplement of B$_\lambda$ if B$_\lambda$G is the full symmetric group on $\Omega$. In [7], Macpherson and Neumann claimed to have classified all supplements of bounded permutation groups. Specifically, they claimed to have proved that a group G acting on the set $\Omega$ is a supplement of B$_\lambda$ if and only if there exists $\Delta \subset \Omega$ with $|\Delta| < \lambda$ such that the setwise stabiliser G$_{\{\Delta\}}$ acts as the full symmetric group on $\Omega\setminus\Delta$. However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory.