diff --git a/django/core/files/base.py b/django/core/files/base.py
index 641ff924c6..85b3c21cba 100644
--- a/django/core/files/base.py
+++ b/django/core/files/base.py
@@ -40,7 +40,7 @@ class File(FileProxyMixin):
         if not hasattr(self, '_size'):
             if hasattr(self.file, 'size'):
                 self._size = self.file.size
-            elif hasattr(self.file, 'name') and os.path.exists(self.file.name):
+            elif hasattr(self.file, 'name') and self.file.name is not None and os.path.exists(self.file.name):
                 self._size = os.path.getsize(self.file.name)
             elif hasattr(self.file, 'tell') and hasattr(self.file, 'seek'):
                 pos = self.file.tell()
diff --git a/tests/files/tests.py b/tests/files/tests.py
index 4f5e171712..be243b2527 100644
--- a/tests/files/tests.py
+++ b/tests/files/tests.py
@@ -205,3 +205,17 @@ class FileMoveSafeTests(unittest.TestCase):
 
         os.close(handle_a)
         os.close(handle_b)
+
+
+class SpooledTempTests(unittest.TestCase):
+    def test_in_memory_spooled_temp(self):
+        with tempfile.SpooledTemporaryFile() as temp:
+            temp.write(b"foo bar baz quux\n")
+            django_file = File(temp, name="something.txt")
+            self.assertEqual(django_file.size, 17)
+
+    def test_written_spooled_temp(self):
+        with tempfile.SpooledTemporaryFile(max_size=4) as temp:
+            temp.write(b"foo bar baz quux\n")
+            django_file = File(temp, name="something.txt")
+            self.assertEqual(django_file.size, 17)