Infrared (IR) and Raman vibrational spectra of materials and surfaces can be rather complex and are often interpreted in combination with simulations based on the double-harmonic approximation. A variety of anharmonic spectral features (band shifts, combination bands, overtones, hot bands, resonances) affect the spectra both quantitatively and qualitatively. We introduce a general method for the efficient description of these anharmonic features---including Fermi and Darling-Dennison resonances---in IR and Raman spectra of materials from first principles. Nuclear quantum effects are explicitly taken into account, and phonon couplings are treated nonperturbatively within a new subspace iterative vibrational configuration interaction (SI-VCI) approach. We apply the SI-VCI method to solid thiourea and solid (dry ice): two molecular crystals with distinct characteristics. Thiourea is a hydrogen-containing system for which we identify a strong three-mode Fermi-like resonance between the fundamental of a symmetric stretching and the combination of two distinct bending modes of groups in the solid phase, which is not observed in the isolated gas-phase molecule. Our results, supported by group theoretical arguments, exclude the presence of bending overtones in the resonant mechanism. Dry ice exhibits four known resonances appearing in the form of dyads and triads (two in the IR spectrum and two in the Raman spectrum), which are also observed in the gas-phase isolated molecule. Our simulations predict an additional Fermi resonance in the IR spectrum of around in the form of a dyad with a splitting . We perform IR transmission experiments where the new resonant spectral feature is observed, assessing the predictiveness of the proposed theoretical approach.