Conventional wisdom has been that the performance limitations in the current Internet lie at the edges of the network -- {\em i.e} last mile connectivity to users, or access links of stub ASes. As these links are upgraded, however, it is important to consider where new bottlenecks and hot-spots are likely to arise. In this paper, we address this question through an investigation of {\em non-access} bottlenecks. These are links within carrier ISPs or between neighboring carriers that could {\em potentially} constrain the bandwidth available to long-lived TCP flows. Through an extensive measurement study, we discover, classify, and characterize bottleneck links (primarily in the U.S.) in terms of their location, latency, and available capacity. We find that about 50\% of the Internet paths explored have a non-access bottleneck with available capacity less than 50 Mbps, many of which limit the performance of well-connected nodes on the Internet today. Surprisingly, the bottlenecks identified are roughly equally split between intra-ISP links and peering links between ISPs. Also, we find that low-latency links, both intra-ISP and peering, have a significant likelihood of constraining available bandwidth. Finally, we discuss the implications of our findings on related issues such as choosing an access provider and optimizing routes through the network. We believe that these results could be valuable in guiding the design of future network services, such as overlay routing, in terms of which links or paths to avoid (and how to avoid them) in order to improve performance.