Doing Bernoulli percolation (each edge is kept with probability $p$, erased otherwise, independently) on a transitive graph yields 0, 1 or infinitely many infinite clusters, and each of these behaviours may occur.
More preciselygenerally, consider a transitive (connected locally finite) graph, i.e. consider a (connected locally finite) graph where all vertices play the same role: formally, we assume that the automorphism group of the graph acts transitively on the vertices. Consider a random subgraph of this graph: you are allowed to randomly erase vertices and edges. Assume that your random process is invariant under the symmetries of your graph: for any automorphism $f$, the distribution of your random subgraph is the same as that of your $f$-translated subgraph. Finally, assume that your process is insertion-tolerant: if $x$ is a vertex or an edge, and if $E$ is an event which does not look at $x$ and which has positive probability, then there is a positive probability that $E$ occurs and $x$ is not erased. Then, almost surely, the number of infinite connected components belongs to $\{0,1,\infty\}$.
(In the case of the so-called Bernoulli percolation, this gives rise to the critical parameters $p_c$ and $p_u$. Another interesting trichotomy in statistical mechanics is subcritical/critical/supercritical; this trichotomy has already been mentioned in this thread, in the setting of PDEs.)