We show that for a suitable class of "Dirac-like" operators there holds a Gluing Theorem for connected sums. More precisely, if M₁ and M₂ are closed Riemannian manifolds of dimension n ≥ 3 together with such operators, then the connected sum M₁ # M₂ can be given a Riemannian metric such that the spectrum of its associated operator is close to the disjoint union of the spectra of the two original operators. As an application, we show that in dimension n = 3 mod 4 harmonic spinors for the Dirac operator of a spin, Spin^c, or Spin^h manifold are not topologically obstructed.