We analyze the performance of the popular class of neighbor-joining methods of phylogeny reconstruction. In particular, we find conditions under which these methods will determine the correct tree topology and show that these perform as well as possible in a certain sense. We also give indications of the performance of these methods when the conditions necessary to show that they determine the entire tree topology correctly, do not hold. We use these results to demonstrate an upper bound on the amount of data necessary to reconstruct the topology with high confidence.