Millimeter Wave (mmWave) networks can deliver multi-Gbps wireless links that use extremely narrow directional beams. This provides us with a new opportunity to exploit spatial reuse in order to scale network throughput. Exploiting such spatial reuse, however, requires aligning the beams of all nodes in a network. Aligning the beams is a difficult process which is complicated by indoor multipath, which can create interference, as well as by the inefficiency of carrier sense at detecting interference in directional links. This paper presents BounceNet, the first many-to-many millimeter wave beam alignment protocol that can exploit dense spatial reuse to allow many links to operate in parallel in a confined space and scale the wireless throughput with the number of clients. Results from three millimeter wave testbeds show that BounceNet can scale the throughput with the number of clients to deliver a total network data rate of more than 39 Gbps for 10 clients, which is up to 6.6× higher than current 802.11 mmWave standards.