||We identify the number of compatibility conditions in 2-D and 3-D discrete structures as a measurement device in determining the rigidity of structures and their resilience to damage. Different shortcuts in counting compatibility conditions are developed and proved. Computing discrete compatibility conditions in hexagonal structures that undergo small enough deformation to be safely linearized, and using these results to say something about the continuum compatibility condition, is the main part of this document. Namely, it is shown that the linearized discrete compatibility condition of hexagonal lattices imposes the 2-D infinitesimal compatibility condition. The computation of discrete 3-D compatibility conditions is done by studying a cuboctahedron. The results show that, because of its infinitesimal flexibility, the extra degree of freedom in a linearized system produces an extra compatibility condition. However, the breaking of nongeneric symmetry of a cuboctahedron by any nonzero perturbation of its nodes makes it infinitesimally rigid, and the extra compatibility condition is lost. The role that compatibility conditions play in damaged structures is demonstrated by experiments on hexagonal lattices. The correlation between the loss of compatibility conditions and the spread of damage is made, as well as the importance of strategic placing of stronger and weaker links in an attempt to strengthen a structure with given boundary conditions and loading.