The goal of this paper is to expose the existence of extremely narrow bandwidth (high Q) resonances in infinitely periodic Frequency Selective Surfaces (FSS) composed of Perfect Electric Conductors (PEC). These narrow bandwidth resonances occur when the current induced on the elements of the FSS has an odd (antisymmetric) distribution. For such resonances of odd current distributions, it is shown that as the incident plane wave approaches normal incidence, the bandwidth approaches 0 Hz (Q approaches ∞). The resonance bandwidth becomes identically 0 (infinite Q) at normal incidence. The currents sustaining such high Q resonances are of very large magnitude. PEC do not dissipate any power, but a small amount of surface resistance of real metals can dissipate enough power (due to the large currents) to completely dampen the high Q resonances. Numerically, as the resonant bandwidth decreases, the condition number of the matrix resulting from the Method of Moments (MoM) formulation increases. For normal incidence (zero bandwidth), the matrix is singular. It is shown that the matrix is ill-posed because of a loss of stability, which comes about in the sense that a small change in the frequency yields a large change in the result.