Metivier proved in 2005 that hyperbolicity, meaning reality of the spectrum of the principal symbol, is a necessary condition for the well-posedness of the initial-value problem associated with first-order quasi-linear operators. Following recent work by Lerner, Morimoto, and Xu, we show that instabilities persist in the limiting case in which hyperbolicity holds initially, but is instantaneously lost. Examples in nonlinear optics (Klein-Gordon-Zakharov) and gas dynamics (Van der Waals, Burgers) illustrate the results. This is joint work with Nicolas Lerner (Paris 6) and Toan Nguyen (Brown).