  The purpose of this paper is twofold. The first is to point out that the
Restricted Isometry Property (RIP) does not hold in many applications where
compressed sensing is successfully used. This includes fields like Magnetic
Resonance Imaging (MRI), Computerized Tomography, Electron Microscopy, Radio
Interferometry and Fluorescence Microscopy. We demonstrate that for natural
compressed sensing matrices involving a level based reconstruction basis (e.g.
wavelets), the number of measurements required to recover all $s$-sparse
signals for reasonable $s$ is excessive. In particular, uniform recovery of all
$s$-sparse signals is quite unrealistic. This realisation shows that the RIP is
insufficient for explaining the success of compressed sensing in various
practical applications. The second purpose of the paper is to introduce a new
framework based on a generalised RIP-like definition that fits the applications
where compressed sensing is used. We show that the shortcomings that show that
uniform recovery is unreasonable no longer apply if we instead ask for
structured recovery that is uniform only within each of the levels. To examine
this phenomenon, a new tool, termed the 'Restricted Isometry Property in
Levels' is described and analysed. Furthermore, we show that with certain
conditions on the Restricted Isometry Property in Levels, a form of uniform
recovery within each level is possible. Finally, we conclude the paper by
providing examples that demonstrate the optimality of the results obtained.
