We study the onset of a yield stress in a polymer microgel dispersion using a combination of particle-tracking microrheology and shear rheometry. On the bulk scale, the dispersion changes from a predominantly viscous fluid to a stiff elastic gel as the concentration of the microgel particles increases. On the microscopic scale, the tracer particles see two distinct microrheological environments over a range of concentrations-one being primarily viscous, the other primarily elastic. The fraction of the material that is elastic on the microscale increases from zero to one as the concentration increases. Our results indicate that the yield stress appears as the result of jamming of the microgel particles, and we infer a model for the small-scale structure and interactions within the dispersion and their relationship to the bulk viscoelastic properties.