Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for acquisition of sparse or compressible signals; instead of taking periodic samples, we measure inner products with M < N random vectors and then recover the signal via a sparsity-seeking optimization or greedy algorithm. Initial research has shown that by leveraging stronger signal models than standard sparsity, the number of measurements required for recovery of an structured sparse signal can be much lower than that of standard recovery. In this paper, we introduce a new framework for structured compressible signals based on the unions of subspaces signal model, along with a new sufficient condition for their recovery that we dub the restricted amplification property (RAmP). The RAmP is the natural counterpart to the restricted isometry property (RIP) of conventional CS. Numerical simulations demonstrate the validity and applicability of our new framework using wavelet-tree compressible signals as an example.