Consider a nonlinear dynamic system affected by possibly concurrent faults (and disturbances). If we wish to detect and isolate each single fault from all other, possibly concurrent faults and disturbances (i.e., to solve the standard Fault Detection and Isolation (FDI) problem), some necessary conditions of differential geometric nature should be satisfied. To deal with cases when these conditions are structurally violated, we introduce a more general class of FDI problems, referred to as Fault Set Detection and Isolation (FSDI), where the focus of the isolation issue is moved from single to sets of faults. Different formulations of FSDI problems are given, necessary and suficient conditions for their solution are provided, and the design of a hybrid residual generator is presented. For the case of non-concurrent faults, it is possible to detect and isolate even single faults under weaker necessary conditions, by processing output residuals that solve suitable FSDI problems. A robotic example is presented to illustrate the theory.