diff --git a/modal/functions.py b/modal/functions.py
index f8ee3ca0c..89289abbe 100644
--- a/modal/functions.py
+++ b/modal/functions.py
@@ -680,11 +680,18 @@ def from_parametrized(
         """mdmd:hidden"""
 
         async def _load(self: _Function, resolver: Resolver, existing_object_id: Optional[str]):
+            try:
+                identity = f"base {self._parent.info.function_name} function"
+            except Exception:
+                # Can't always look up the function name that way, so fall back to generic message
+                identity = "base function for parameterized class"
             if not self._parent.is_hydrated:
+                if self._parent.app._running_app is None:
+                    reason = ", because the App it is defined on is not running."
+                else:
+                    reason = ""
                 raise ExecutionError(
-                    "Base function in class has not been hydrated. This might happen if an object is"
-                    " defined on a different stub, or if it's on the same stub but it didn't get"
-                    " created because it wasn't defined in global scope."
+                    f"The {identity} has not been hydrated with the metadata it needs to run on Modal{reason}."
                 )
             assert self._parent._client.stub
             serialized_params = serialize((args, kwargs))
diff --git a/modal/object.py b/modal/object.py
index d2c047c10..26f7c9b45 100644
--- a/modal/object.py
+++ b/modal/object.py
@@ -204,11 +204,15 @@ async def resolve(self):
         if self._is_hydrated:
             return
         elif not self._hydrate_lazily:
+            object_type = self.__class__.__name__.strip("_")
+            if hasattr(self, "_app") and getattr(self._app, "_running_app", "") is None:
+                # The most common cause of this error: e.g., user called a Function without using App.run()
+                reason = ", because the App it is defined on is not running."
+            else:
+                # Technically possible, but with an ambiguous cause.
+                reason = ""
             raise ExecutionError(
-                "Object has not been hydrated and doesn't support lazy hydration."
-                " This might happen if an object is defined on a different stub,"
-                " or if it's on the same stub but it didn't get created because it"
-                " wasn't defined in global scope."
+                f"{object_type} has not been hydrated with the metadata it needs to run on Modal{reason}."
             )
         else:
             # TODO: this client and/or resolver can't be changed by a caller to X.from_name()