This technical note investigates the robust stability of uncertain linear time-varying systems with a mode-switch dynamics. Each mode is characterized by a dynamical matrix containing elements whose time behavior over bounded time intervals is described by interval polynomials of arbitrary degree. Using a quadratic Lyapunov function polynomially depending on time, stability conditions for each mode are stated in terms of linear matrix inequalities (LMIs). The stability conditions of the switching system are stated both in terms of minimum and average dwell time. A salient feature of the technical note is that the single-mode stability conditions given here allow the parameters and their derivatives to take values over arbitrarily large uncertainty sets.