Under certain circumstances the excitations of a superconductor will not be those predicted by BCS theory, but rather electron bag states. A variational calculation is used to establish this regime in which such collective states form. The fact that such bags may bind more than one electron implies the existence of more than one peak in the plots of dI/dV versus VP obtained in Giaever tunneling experiments. The estimate of the number and ratios of voltages associated with such peaks is in qualitative agreement with recent experimental results.