We consider zero range processes with density dependent jump rates g given by $g=g(n,k)=g_1(n)g_2(k/n)$ with $g_1(x)=x^{-alpha}$ and $$g_2(x)={^{x^{-alpha};if;a<x}_{Mx^{-alpha};if;x{leq}a}$$. (0.1) In this case, with 1/2 < a < 1 and ${alpha}$ > 0, we show that non-complete condensation occurs with maximum cluster size an. More precisely, for any ${epsilon}$ > 0, there exists $M^*$ > 0 such that, for any 0 < M ${leq}M^*$, the maximum cluster size is between (a - ${epsilon}$)n and (a + ${epsilon}$)n for large n. This provides a simple example of non-complete condensation under perturbation of rates which are deep in the range of perfect condensation (e.g. ${alpha}$ >> 1) and supports the instability of the condensation transition.