Steady rectilinear motion of incompressible non-Newtonian fluids is, in general, characterized by the two planar fields of pressure and speed, and these are in turn restricted by the three differential equations of motion. Two material functions which depend on the shear rate and which are associated with normal and shear stress effects in viscometric flows enter these equations in such a way that if the material functions are not proportional, the three field equations are independent. For this prevalent case of apparent overdetermination, a well-known and generally accepted conjecture of Ericksen (1956) severely delimits the class of possible flows. In fact, from his conjecture Ericksen drew the remarkable and profound secondary conclusion that rectilinear steady flow through a cylindrical pipe to which the fluid adheres is impossible, except perhaps when the cylinder is composed of portions of planes and right circular cylinders. In this paper, while we show that Ericksen's conjecture is not fully valid, we also find that in cases of major importance his secondary conclusion is correct in even a sharper form. Our main result (stated in the Introduction) shows that subject to certain technical requirements, rectilinear steady flow with boundary adherence through a fixed straight pipe is possible only when the pipe is either circular or the annulus between two concentric circles. Counter-examples are given which illustrate that the independence condition alone is not sufficient to justify either Ericksen's conjecture, his secondary conclusion, or our main result. Finally, we show that our main result is valid when the condition of boundary adherence is replaced by a natural slip boundary condition.