We calculate the heat-kernel coefficients, up to a(2), for a U(1) bund
le on the 4-ball for boundary conditions which are such that the norma
l derivative of the held at the boundary is related to a first-order o
perator in boundary derivatives acting on the held. The results are us
ed to place restrictions on the general forms of the coefficients. In
the specific case considered, there can be a breakdown of ellipticity.