We present a general $S$-matrix approach to the phenomenological treatment of scattering problems which involve many overlapping resonances in the presence of a constant background. The problem is treated systematically using the analyticity and unitarity of $S(E)$, and $\mathrm{CPT}$ and $T$ invariance. We derive the possible factorization properties of the resonance residues for first-order and higher-order poles, the restriction imposed by $\mathrm{CPT}$ invariance, the restrictions imposed by $\mathrm{CPT}$ and $T$, and the constraints on the residues imposed by unitarity. The nature of the contraints is investigated in detail for overlapping first-order resonances, and in less detail for higher-order resonances. The results are illustrated by several examples, including the case of a single dipole resonance in the presence of background.