The best-effort nature of the Internet poses a significant obstacle to the deployment of many applications that require guaranteed bandwidth. We present a novel approach that enables two edge/border routers which we call Internet traffic managers (ITM) - to use an adaptive number of TCP connections to set up a tunnel of desirable bandwidth between them. The number of TCP connections that comprise this tunnel is elastic in the sense that it increases/decreases in tandem with competing cross traffic to maintain a target bandwidth. An origin ITM would then schedule incoming packets from an application requiring guaranteed bandwidth over that elastic tunnel. Unlike many proposed solutions that aim to deliver soft QoS guarantees, our elastic-tunnel approach does not require any support from core routers (as with IntServ and DiffServ); it is scalable in the sense that core routers do not have to maintain per-flow state (as with IntServ); and it is readily deployable within a single ISP or across multiple ISPs. To evaluate our approach, we develop a flow-level control theoretic model to study the transient behavior of established elastic TCP-based tunnels. The model captures the effect of cross-traffic connections on our bandwidth allocation policies. Through extensive simulations, we confirm the effectiveness of our approach in providing soft bandwidth guarantees.