Consider the events $\(F_n \cap \bigcup_(k=1)^(n-1) F_k = \emptyset\)$, $n \in \bN$, where $(F_n(n=1)^\infty$ is an i.i.d. sequence of stationary random subsets of a compact group $\bG$. A plausible conjecture is that these events will not occur infinitely often with positive probability if $\bP\(F_i \cap F_j \ne \emptyset \, | \, F_j\) > 0$ a.s. for $i \ne j$. We present a counterexample to show that this condition is not sufficient, and give one that is. The sufficient condition always holds when $F_n = \(X_t^n : 0 \le t \le T\)$ is the range of a L\'evy process $X^n$ on the $d$-dimensional torus with uniformly distributed initial position and $\bP\(\exists 0 \le s, t \le T : X_s^i = X_t^j \) > 0$ for $i \ne j$. We also establish an analogous result for the sequence of graphs $\((t,X_t^n) : 0 \le t \le T\)$.