An old result assigned to B. H. Neumann by \textit{L. Fuchs} [in Infinite Abelian groups. Vol. II (1973; Zbl 0257.20035)] states that if \(S_1,\dots,S_n\) are proper subgroups of an Abelian group \(A\) such that \(A=\bigcup_{i=1}^n(a_i+S_i)\) then one of the subgroups \(S_i\) is a finite index subgroup of \(A\). The author shows that (i) if \(n\geq 3\) then one of these subgroups has index at most \(2^{2(n-2)}-2^{n-2}+1\), and (ii) if \(n\geq 4\) then there exist two subgroups \(S_i,S_j\) of index at most \(2^{2(n-2)}-2^{n-2}+1\) (Theorem 3). When \(n=3\) it is shown that the property (i) is not valid for non-Abelian groups and the property (ii) is not valid even for Abelian groups.