We start with BPS-saturated configurations of two (orthogonally) intersecting M-branes and use the electro-magnetic duality or dimensional reduction along a boost, in order to obtain new p-brane bound states. In the first case the resulting configurations are interpreted as BPS-saturated non-threshold bound states of intersecting p-branes, and in the second case as p-branes intersecting at angles and their duals. As a by-product we deduce the enhancement of supersymmetry as the angle approaches zero. We also comment on the D-brane theory describing these new bound states, and a connection between the angle and the world-volume gauge fields of the D-brane system. We use these configurations to find new embeddings of the four and five dimensional black holes with non-zero entropy, whose entropy now also depends on the angle and world-volume gauge fields. The corresponding D-brane configuration sheds light on the microscopic entropy of such black holes.