Matrices of nonnegative rank at most r form a semialgebraic set. This semialgebraic set is understood for r=1,2,3. In this talk, I will recall what was previously known about this semialgebraic set and present recent results on the boundaries of the set of matrices of nonnegative rank at most four using notions from the rigidity theory of frameworks. These results are joint work with Robert Krone. In the nonnegative rank three case, all boundaries are trivial or consist of matrices that have only infinitesimally rigid factorizations. For arbitrary nonnegative rank, we give a necessary condition on zero entries of a nonnegative factorization for the factorization to be infinitesimally rigid, and we show that in the case of 5×5 matrices of nonnegative rank four, there exists an infinitesimally rigid realization for every zero pattern that satisfies this necessary condition. However, the nonnegative rank four case is much more complicated than the nonnegative rank three case, and there exist matrices on the boundary that have factorizations that are not infinitesimally rigid. We discuss two such examples.