The perturbations induced by the varying current I or magnetic field may result in the instability of the superconducting state at some I= I m < I s , where I m is the function of dI dt . It is shown that cooperative effect of the instability and the normal zone propagation may be responsible for anomalies of quench processes. For example, the propagation velocity increases sharply and the quench energy tends to zero at I∼ I m since at I > I m there is no stable homogeneous superconducting state. The obtained results may be essential for understanding of the quench in the pulse magnet systems (such as SMES) and sectional magnets where normal transition accomplished by the fast current exchange between the sections.