Finding the nearest peer, in terms of latency, is an important problem in many Internet applications. In this paper, we argue that solutions that only examine inter-peer latencies as part of their operation will find it infeasible, in certain cases, to discover the nearest peer in P2P systems. The difficulty arises out of the way the last hop is typically laid out in the Internet, where a single PoP (point of presence) belonging to an ISP provides connectivity to numerous client networks. This setup leads to a large number of peers being at about the same latency from one another, which, we argue, presents a serious obstacle when a peer tries to discover another peer residing in the same network as itself. We use large-scale measurements over hosts in the Azureus P2P network and DNS servers to show that this condition does occur in real settings, and use simulations of the Meridian closest-server algorithm to show that the condition does indeed lead to difficulty in finding the exact-closest peer. We also propose a few heuristics that help address this issue.