Summary: The method of minimum evolution reconstructs a phylogenetic tree \(T\) for \(n\) taxa given dissimilarity data \(d\). In principle, for every tree \(W\) with these \(n\) leaves an estimate for the total length of \(W\) is made, and \(T\) is selected as the \(W\) that yields the minimum total length. Suppose that the ordinary least-squares formula \(S_W(d)\) is used to estimate the total length of \(W\). A theorem of Rzhetsky and Nei shows that when \(d\) is positively additive on a completely resolved tree \(T\), then for all \(W \neq T\) it will be true that \(S_W(d) > S_T(d)\). The same will be true if \(d\) is merely sufficiently close to an additive dissimilarity function. This paper proves that as \(n\) grows large, even if the shortest branch length in the true tree \(T\) remains constant and \(d\) is additive on \(T\), then the difference \(S_ W(d)-S_T(d)\) can go to zero. It is also proved that, as \(n\) grows large, there is a tree \(T\) with \(n\) leaves, an additive distance function \(d_T\) on \(T\) with shortest edge \(\epsilon\), a distance function \(d\), and a tree \(W\) with the same \(n\) leaves such that \(d\) differs from \(d_T\) by only approximately \(\epsilon/4\), yet minimum evolution incorrectly selects the tree \(W\) over the tree \(T\). This result contrasts with the method of neighbor-joining, for which Atteson showed that incorrect selection of \(W\) required a deviation at least \(\epsilon/2\). It follows that, for large \(n\), minimum evolution with ordinary least-squares can be only half as robust as neighbor-joining.