How does Camerini's algorithm for minimum-bottleneck-spanning-tree run in linear time?
First, after each stage throw away any isolated vertices. With these vertices removed (even when the graph is disconnected) the number of vertices in the next stage will be at most twice the number of edges.

Next, use the fact that (with isolated vertices removed, in each stage after the first) $|V|\le 2|E|$ to simplify the time bound in each stage (after the first) from $O(|V|+|E|)$ to $O(|E|)$.

Finally, observe that at each stage $|E|$ goes down by a factor of two, so all these $O(|E|)$ bounds add in a geometric series to be linear in the size of the original graph.