Compressed sensing is a theory which guarantees the exact recovery of sparse signals from a small number of linear projections. The sampling schemes suggested by current compressed sensing theories are often of little practical relevance since they cannot be implemented on real acquisition systems. In this paper, we study a new random sampling approach that consists in projecting the signal over blocks of sensing vectors. A typical example is the case of blocks made of horizontal lines in the 2D Fourier plane. We provide theoretical results on the number of blocks that are required for exact sparse signal reconstruction. This number depends on two properties named intra and inter-support block coherence. We then show through a series of examples including Gaussian measurements, isolated measurements or blocks in time-frequency bases, that the main result is sharp in the sense that the minimum amount of blocks necessary to reconstruct sparse signals cannot be improved up to a multiplicative logarithmic factor. The proposed results provide a good insight on the possibilities and limits of block compressed sensing in imaging devices such as magnetic resonance imaging, radio-interferometry or ultra-sound imaging.