Bootstrap percolation is a well-known class of monotone cellular automata, in which sites may be infected or not and, at any step, a site becomes infected if a certain constraint is satisfied. Bootstrap percolation has a non-monotone stochastic counterpart , kinetically constrained models (KCM), which were introduced to model the liquid/glass transition, a major open problem of condensed matter physics. In KCM, the state of each site is re-sampled (independently) at rate 1 if the constraint is satisfied. A key problem for KCM is to determine the divergence of timescales as p → 0, where p is the equilibrium density of infected sites. In this article we establish a combinatorial result which in turn allows to prove a lower bound on timescales for KCM.