An analysis is presented of spherical and cylindrical sink flows of viscoelastic fluids. Stress levels and the rate of stress growth are found to be restricted to modest magnitudes in arbitrarily rapid cylindrical sink flows (flows through a wedge), but in contrast, stresses rise very rapidly to high levels during flows through conical channels. The analysis appears to be in qualitative agreement with the experimental results observed by James and Saringer for very dilute polymer solutions and is presented as an interpretation of those exceptional data. Consequences of the major differences between the two flow fields are explored briefly, and the range of applicability of the analysis is considered.