When using Heroku PGBackups (ref: https://devcenter.heroku.com/articles/heroku-postgres-backups) to backup or copy across a database, you receive the error message "Could not connect to database to initialize transfer".
In the majority of cases, this is caused by the database connections being maxed out, for example, having 20/20 connections reported for a basic (formerly hobby-basic) tier Heroku Postgres database. This can be checked using
=== DATABASE_URL Plan: Hobby-basic Status: Available Connections: 20/20 PG Version: 10.6 Created: 2018-11-12 02:23 UTC Data Size: 11.5 MB Tables: 17 Rows: 10000/10000000 (In compliance) Fork/Follow: Unsupported Rollback: Unsupported Continuous Protection: Off Add-on: postgresql-adjective-12345
In order for the backup process to succeed, Heroku must be able to connect to the database. As a result, you will need to put the application into maintenance mode (ref: https://devcenter.heroku.com/articles/maintenance-mode) and/or scale down dynos to reduce the number of connections to your Heroku Postgres database.