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We show that the commonly used lowest-order theory of phonon-phonon interactions frequently fails to accurately describe the anharmonic phonon decay rates and thermal conductivity ($κ$), even among strongly bonded crystals. Applying a first principles theory that includes both the lowest-order three-phonon and the higher-order four-phonon processes to seventeen zinc blende semiconductors, we find that the lowest-order theory drastically overestimates the measured $κ$ for many of these materials, while inclusion of four-phonon scattering gives significantly improved agreement with measurements. We have identified new selection rules on three-phonon processes that help explain many of these failures in terms of anomalously weak anharmonic phonon decay rates predicted by the lowest-order theory competing with four-phonon processes. We also show that zinc blende compounds containing boron (B), carbon (C) or nitrogen (N) atoms have exceptionally weak four-phonon scattering, much weaker than in compounds that do not contain B, C or N atoms. This new understanding helps explain the ultrahigh $κ$ in several technologically important materials like cubic boron arsenide, boron phosphide and silicon carbide. At the same time, it not only makes the possibility of achieving high $κ$ in materials without B, C or N atoms unlikely, but it also suggests that it may be necessary to include four-phonon processes in many future studies. Our work gives new insights into the nature of anharmonic processes in solids and demonstrates the broad importance of higher-order phonon-phonon interactions in assessing the thermal properties of materials.