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Many sparse approximation algorithms accurately recover the sparsest solution to an underdetermined system of equations provided the matrix's restricted isometry constants (RICs) satisfy certain bounds. There are no known large deterministic matrices that satisfy the desired RIC bounds; however, members of many random matrix ensembles typically satisfy RIC bounds. This experience with random matrices has colored the view of the RICs' behavior. By modifying matrices assumed to have bounded RICs, we construct matrices whose RICs behave in a markedly different fashion than the classical random matrices; RICs can satisfy desirable bounds and also take on values in a narrow range.