•Models with no solution are incoherent; with multiple solutions are incomplete.•Models with occasionally binding constraints are not generically coherent.•Coherency needs restrictions on parameters or on the support of shock distributions.•These restrictions generally cause incompleteness.•The NK model is incoherent under typical parameterizations, but UMP can help. Structural models with no solution are incoherent, and those with multiple solutions are incomplete. We show that models with occasionally binding constraints are not generically coherent. Coherency requires restrictions on the parameters or on the support of the distribution of the shocks. In presence of multiple shocks, the support restrictions cannot be independent from each other, so the assumption of orthogonality of structural shocks is incompatible with coherency. Models whose coherency is based on support restrictions are generically incomplete, admitting a very large number of minimum state variable solutions.