Low Reynolds number swimming of microorganisms in Newtonian fluids is an extensively studied classical problem.  However, many biological fluids such as mucus are mixtures of water and polymers and are more appropriately described as viscoelastic fluids. Recently, there have been many studies on locomotion in complex fluids. Both experiments and theory have exhibited that viscoelasticity can lead to either an enhancement or retardation of swimming, but a complete understanding of this problem is lacking. A computational model of finite-length undulatory swimmers is used to examine the physical origin of the effect of elasticity on swimming speed.  We reproduce conflicting results from the literature simply by changing relevant physical parameters.  Additionally, we examine an oscillatory bending beam in a viscoelastic fluid, and identify a threshold in amplitude related to the development of large elastic stresses like those observed in the swimmer problem which are involved in the elastic enhancement.  We relate this transition to previously studied bifurcations in steady extensional flows of complex fluids.  This reduced model sheds some light on properties of swimmer gaits that lead to either elastic enhancement or hindrance.