The time-strain separability in viscoelastic systems is not a rule derived from fundamental principles but merely a hypothesis based on experimental phenomena, stress relaxation for long times. The violation of separability in the short-time response just after step strain is also well understood [L. A. Archer, J. Rheol. 43, 1555 (1999)]. In constitutive modeling, time-strain separability has been extensively employed because of its theoretical simplicity and practical convenience. Here we present a simple analysis that verifies this hypothesis inevitably incurs mathematical inconsistency from the viewpoint of stability. Employing an asymptotic analysis, we show that both differential and integral constitutive equations based on time-strain separability are either Hadamard type unstable or dissipative unstable. It is shown that the Hadamard-type instabilities of the Wagner, Luo-Tanner, Papanastasiou, and Kaye-Bernstein-Kearsley-Zapas models with Larson-Monroe or Mooney potential, as well as the dissipative instability of the Lodge model (all proven previously) [Y. Kwon and A. I. Leonov, Rheol. Acta 33, 398 (1995)] are all caused by the separability hypothesis inherent in their equations. The conclusion drawn in this study is shown to be applicable to the Doi-Edwards model (with independent alignment approximation). Hence, the Hadamard-type instability of the Doi-Edwards model results from the time-strain separability in its formulation and its remedy may lie in the transition mechanism from Rouse to reptational relaxation suggested by Doi and Edwards.