In this paper we study a one-dimensional (1-D) quasilinear hyperbolic model for the elongation and onset of failure of polymeric liquid filaments. Our model is derived from the full 3-D free surface boundary value problem, assuming a Maxwell constitutive law for the polymeric liquid, by (a) positing a dominant balance of inertia, surface tension, gravity, viscosity and elastic relaxation, and (b) applying the slender axisymmetric perturbation methods of Bechtel et al., J. Stability Appl. Anal. Continua, 2 (1992) 1. The transient 1-D approximate equations in Lagrangian form consist of a coupled system of five, first-order quasilinear partial differential equations which admit explicit characteristic analysis and classification. The system may be genuinely hyperbolic or mixed elliptic-hyperbolic type. Importantly, a simulation which begins hyperbolic may evolve into the ill-posed mixed type regime at a particular time and location. Such a transition corresponds to a departure from slender behavior and breakdown of the 1-D theory, which we equate with onset of filament failure. The detailed nature of this transition from slender behavior is given particular scrutiny here. Our 1-D model is applied with appropriate initial and boundary data to simulate the experimental condition＇s in which one end of a filament is attached to a ceiling and the other to a falling drop which pulls a slender filament behind it. We show a representative sample of extensive parameter studies in which the model for the filament is initially a well-posed hyperbolic system, and numerical predictions are made on the filament behavior as it stretches. A remarkable variety of filament behavior is predicted within this 1-D Maxwell model. As we vary the competing physical effects, we predict qualitatively different locations, timescales and modes of onset of failure.