diff --git a/django/db/backends/postgresql/base.py b/django/db/backends/postgresql/base.py
index 7926c05c09..126f30b462 100644
--- a/django/db/backends/postgresql/base.py
+++ b/django/db/backends/postgresql/base.py
@@ -332,6 +332,9 @@ class DatabaseWrapper(BaseDatabaseWrapper):
                             yield cursor
                     finally:
                         conn.close()
+                    break
+            else:
+                raise
 
     @cached_property
     def pg_version(self):
diff --git a/docs/releases/3.1.7.txt b/docs/releases/3.1.7.txt
index f6be18b0e8..1ef16b76c7 100644
--- a/docs/releases/3.1.7.txt
+++ b/docs/releases/3.1.7.txt
@@ -9,4 +9,6 @@ Django 3.1.7 fixes several bugs in 3.1.6.
 Bugfixes
 ========
 
-* ...
+* Fixed a regression in Django 3.1 that caused ``RuntimeError`` instead of
+  connection errors when using only the ``'postgres'`` database
+  (:ticket:`32403`).
diff --git a/tests/backends/postgresql/tests.py b/tests/backends/postgresql/tests.py
index 94ebfbdaf8..7f27c69ffb 100644
--- a/tests/backends/postgresql/tests.py
+++ b/tests/backends/postgresql/tests.py
@@ -1,9 +1,10 @@
+import copy
 import unittest
 from io import StringIO
 from unittest import mock
 
 from django.core.exceptions import ImproperlyConfigured
-from django.db import DatabaseError, connection, connections
+from django.db import DEFAULT_DB_ALIAS, DatabaseError, connection, connections
 from django.db.backends.base.base import BaseDatabaseWrapper
 from django.test import TestCase, override_settings
 
@@ -54,6 +55,53 @@ class Tests(TestCase):
         self.assertIsNone(cursor.db.connection)
         self.assertIsNotNone(cursor.db.settings_dict['NAME'])
         self.assertEqual(cursor.db.settings_dict['NAME'], connections['other'].settings_dict['NAME'])
+        # Cursor is yielded only for the first PostgreSQL database.
+        with self.assertWarnsMessage(RuntimeWarning, msg):
+            with mock.patch(
+                'django.db.backends.base.base.BaseDatabaseWrapper.connect',
+                side_effect=mocked_connect,
+                autospec=True,
+            ):
+                with connection._nodb_cursor() as cursor:
+                    self.assertIs(cursor.closed, False)
+                    self.assertIsNotNone(cursor.db.connection)
+
+    def test_nodb_cursor_raises_postgres_authentication_failure(self):
+        """
+        _nodb_cursor() re-raises authentication failure to the 'postgres' db
+        when other connection to the PostgreSQL database isn't available.
+        """
+        def mocked_connect(self):
+            raise DatabaseError()
+
+        def mocked_all(self):
+            test_connection = copy.copy(connections[DEFAULT_DB_ALIAS])
+            test_connection.settings_dict = copy.deepcopy(connection.settings_dict)
+            test_connection.settings_dict['NAME'] = 'postgres'
+            return [test_connection]
+
+        msg = (
+            "Normally Django will use a connection to the 'postgres' database "
+            "to avoid running initialization queries against the production "
+            "database when it's not needed (for example, when running tests). "
+            "Django was unable to create a connection to the 'postgres' "
+            "database and will use the first PostgreSQL database instead."
+        )
+        with self.assertWarnsMessage(RuntimeWarning, msg):
+            mocker_connections_all = mock.patch(
+                'django.utils.connection.BaseConnectionHandler.all',
+                side_effect=mocked_all,
+                autospec=True,
+            )
+            mocker_connect = mock.patch(
+                'django.db.backends.base.base.BaseDatabaseWrapper.connect',
+                side_effect=mocked_connect,
+                autospec=True,
+            )
+            with mocker_connections_all, mocker_connect:
+                with self.assertRaises(DatabaseError):
+                    with connection._nodb_cursor():
+                        pass
 
     def test_database_name_too_long(self):
         from django.db.backends.postgresql.base import DatabaseWrapper