diff --git a/src/flask/app.py b/src/flask/app.py
index 1143f20d..3b9d062d 100644
--- a/src/flask/app.py
+++ b/src/flask/app.py
@@ -1270,7 +1270,9 @@ class Flask(_PackageBoundObject):
         else:
             exc_class = exc_class_or_code
 
-        assert issubclass(exc_class, Exception)
+        assert issubclass(
+            exc_class, Exception
+        ), "Custom exceptions must be subclasses of Exception."
 
         if issubclass(exc_class, HTTPException):
             return exc_class, exc_class.code
diff --git a/tests/test_user_error_handler.py b/tests/test_user_error_handler.py
index cd2b5864..d9f94a3f 100644
--- a/tests/test_user_error_handler.py
+++ b/tests/test_user_error_handler.py
@@ -11,11 +11,19 @@ def test_error_handler_no_match(app, client):
     class CustomException(Exception):
         pass
 
+    class UnacceptableCustomException(BaseException):
+        pass
+
     @app.errorhandler(CustomException)
     def custom_exception_handler(e):
         assert isinstance(e, CustomException)
         return "custom"
 
+    with pytest.raises(
+        AssertionError, match="Custom exceptions must be subclasses of Exception."
+    ):
+        app.register_error_handler(UnacceptableCustomException, None)
+
     @app.errorhandler(500)
     def handle_500(e):
         assert isinstance(e, InternalServerError)