Postgres: FATAL too many clients already

What the message looks like

FATAL: sorry, too many clients already
connection to server at "db" (10.0.0.5), port 5432 failed

The application opens connections and never closes them, for example after a deploy or in a worker without finally.
The sum of all pool sizes exceeds max_connections, especially with several instances or containers.
A PgBouncer in session mode passes every client connection through and defeats pooling.
Transactions stay idle in transaction and hold slots without doing work.
Backup, monitoring and migration jobs add to the application pools.

Used and allowed connections
psql -c "SELECT count(*) AS used, current_setting('max_connections')::int AS max FROM pg_stat_activity;"
Who holds the connections?
psql -c "SELECT usename, state, count(*) FROM pg_stat_activity GROUP BY 1,2 ORDER BY 3 DESC;"
Long idle transactions
psql -c "SELECT pid, now()-state_change AS idle_for, query FROM pg_stat_activity WHERE state LIKE 'idle in transaction%' ORDER BY 2 DESC LIMIT 10;"
Reserved slots
psql -c "SHOW superuser_reserved_connections;"

Measure first, then raise: a bigger max_connections only hides a leak and costs memory per backend.
Close the leak: use context managers or finally, and keep each pool smaller than max_connections / instances.
For many short connections put PgBouncer in transaction mode in front.
Set idle_in_transaction_session_timeout so stuck transactions are terminated.
Raise the limit only with memory headroom for autovacuum and backups, and remember it needs a restart.

Briefly, yes. With a leak the error returns within hours. Measure who holds the slots first.

Reserved slots for superusers and background processes count against the limit, so a normal client hits it earlier.