Solution NMR peak widths are mostly determined by two factors, homogeneous broadening and inhomogeneous broadening. Homogeneous broadening is governed by T₂ relaxation, which is driven by the dynamics of the molecular segments. Small molecules rotate very fast in solution (>> 10⁹ s^-1), resulting in long T₂ (~ seconds) and thus very sharp peaks (intrinsic peak width usually < 0.5 Hz). For polymers, there are different situations. For hydrocarbon polymers, each bond of the backbone rotates independent of the bonds that are more than 4-5 bonds away, and such rotations are almost as fast as those of small molecules. Therefore, NMR peaks of hydrocarbon polymers are almost as sharp as those of small molecules, and the peak widths are largely independent of molecular weight. For polymers that have strong interactions with solvents, the bond rotations might not be that free, resulting in shorter T₂ and thus broader peak widths. You could think of the former case as a snake freely swirling in water while the latter case as a stiff coil of steel wire.

Inhomogeneous broadening is governed by the heterogeneity of chemical structure of the molecule. For example, a purely isotactic polymer has a uniform chemical structure throughout the backbone, while an atactic polymer has many different local structures (mmr, mrm, mrr, etc.). Therefore, an isotactic polystyrene has very sharp NMR peaks, while an atactic polystyrene has quite broad NMR peaks. The difference here is not homogeneous broadening due to T₂ relaxation, but due to inhomogeneous broadening due to tacticity. In other words, the broad peaks on polystyrene NMR spectra are in fact many sharp peaks right next to each other.

In summary, polymer peaks are not always broad, and shimming is generally recommended for polymer samples. In fact, the extent of the peak broadening tells you a lot about the behavior of these molecules and their interaction with their surrounding environment.