Even a large confluence of connections like this would likely still have had local switchboards. When you made a call, your local operator would have connected

Even a large confluence of connections like this would likely still have had local switchboards. When you made a call, your local operator would have connected you either to a local number on their own board, or to another local board as needed. That second operator would have then connected you to the desired number. Each board would only have limited connection lines to each other board (or to a branch exchange). So if all the connections from board A to board B (or to the branch exchange) were in use, the caller would have to try again later.