In this paper, we explore multi-hop bandwidth estimation assuming arbitrary cross-traffic at each node and develop a new probing method called Envelope that can asymptotically estimate not only the available bandwidth but also the raw capacity of the tight link. Envelope is based on a multi-link recursive extension of unbiased single-hop estimators proposed in the past (e.g., [14]) and a variation of the packet-cartouche technique [6]. Through extensive simulations, we evaluate Envelope in various network settings and cross-traffic conditions and find that it can measure tight-link bandwidth characteristics with accuracy that significantly surpasses that of the existing methods. We also find that Envelope can measure non-tight links in certain path and cross-traffic configurations.