The one‐dimensional equations describing Newtonian fiber flow have been derived previously using lubrication scaling of the axisymmetric equations of motion. Here, we present an extension of that analysis using a generalized convected Maxwell model. We find that the non‐Newtonian behavior of the fluid must be severely limited for the one‐dimensional equations to be determinate. In this sense, the primary effect of viscoelastic behavior is to make the flow more two dimensional. Limiting cases based on previous one‐dimensional models are shown to violate the axisymmetric equations of motion. In addition, questions are raised about the validity of elongational viscosity measurements of highly elastic fluids.