Quite late to the party here, but I ran out of database connections today in production.
Like a lot of people, I use Sidekiq to perform asynchronous jobs like sending emails for example. It is important to note that Sidekiq runs as a multithread process.
So, I don’t just have a single-threaded Rails application, therefore this answer does not directly apply to the question asked but I thought it was worth saying something here as I think multithreaded Rails apps are relatively normal nowadays.
This means you need to adjust your pool size in such a way as to create enough connections to handle all the jobs that can be enqueued and take longer than 5 seconds (the default timeout period to wait for a database connection before throwing an error).