The production rate of polymer fibers by extrusion is usually limited by the appearance of a series of instabilities ("melt fracture") that lead to unwanted undulations of the surface. We present both qualitative and quantitative experimental evidence that-in addition to previously known polymer-specific scenarios-there is an intrinsic route towards melt fracture type phenomena: a nonlinear ("subcritical") instability of viscoelastic Poiseuille flow.