Connection/Pooling Issue with Cloud SQL

I've gotten my app and my database talking to each other, but my SQL and App Engine logs are being absolutely flooded by connection errors anyway.

On the App Engine side, I'm seeing the usual `/cloudsql/project:region:database/.s.PSQL.5432` not found error. But on the Cloud SQL side, I'm seeing `FATAL: remaining connection slots are reserved for non-replication superuser connections`

Again: I have pages that launch DB queries that I can load reliably without 500s, so they are actually talking to each other, but something else is clearly going on here.

My limited research has shown that a base PostgresQL instance can support up to 100 connections, and the default pool size I've seen for Phoenix in production is 20.