Data on the parallel pump and subsidiary absorption threshold at 9.1 GHz and effective linewidth &Dgr;Heffat 9.9 GHz have been obtained for polycrystalline YIG materials with an average grain diametera0from 1 to 30 &mgr;. The fine‐grain samples (1–10 &mgr;) were made by hot pressing. The parallel pump spin‐wave line‐widths extrapolated to zero wavenumber, &Dgr;Hk→0, vary asa0−1, with &Dgr;Hk→0=25 Oe ata0=1 &mgr;. The &Dgr;Hkfor the fine‐grain samples (a0≈ 1 &mgr;) show an inversek‐dependence. A simple model in which the spin‐wave transit time across individual grains limits &Dgr;Hkappears to explain thea0−1dependence but not the observedk‐dependences for &Dgr;Hk. The high‐field &Dgr;Heffis 5.5 Oe fora0=1 &mgr; and varies asa0−1/2. The low‐field values are slightly larger, due to a small two‐magnon contribution arising from coupling to degenerate short‐wavelength spin waves. Even though the high‐field &Dgr;Heffcontains no two‐magnon contribution, it is significantly larger than linewidths in single crystals. A recently proposed magnetostriction mechanism is consistent with thea0−1/2dependence, but the predicted magnitude of &Dgr;Heffis much smaller than observed.