We study the constraints on models with extra dimensions arising from local anomaly cancellation. We consider a five-dimensional field theory with a U(I) gauge field and a charged fermion, compactified on the orbifold S-1/(Z(2) x Z(2)'). We show that, even if the orbifold projections remove both fermionic zero modes, there are gauge anomalies localized at the fixed points. Anomalies naively cancel after integration over the fifth dimension, but gauge invariance is broken, spoiling the consistency of the theory. We discuss their implications for realistic supersymmetric models with a single Higgs hypermultiplet in the bulk, and possible cancellation mechanisms in non-minimal models. (C) 2002 Elsevier Science B.V. All rights reserved.