added test

src/textual/app.py

@@ -1386,11 +1386,10 @@ class App(Generic[ReturnType], DOMNode):
                 raise
 
             finally:
+                self._running = True
                 await self._ready()
                 await invoke_ready_callback()
 
-            self._running = True
-
             try:
                 await self._process_messages_loop()
             except asyncio.CancelledError:

tests/test_binding_inheritance.py

@@ -11,13 +11,12 @@ background relating to this.
 
 from __future__ import annotations
 
-import pytest
-
 from textual.app import App, ComposeResult
-from textual.widgets import Static
-from textual.screen import Screen
 from textual.binding import Binding
 from textual.containers import Container
+from textual.screen import Screen
+from textual.widget import Widget
+from textual.widgets import Static
 
 ##############################################################################
 # These are the movement keys within Textual; they kind of have a special
@@ -574,6 +573,7 @@ class PriorityOverlapScreen(Screen):
     def on_mount(self) -> None:
         self.query_one(PriorityOverlapWidget).focus()
 
+
 class PriorityOverlapApp(AppKeyRecorder):
     """An application with a priority binding."""
 
@@ -607,3 +607,40 @@ async def test_overlapping_priority_bindings() -> None:
             "app_e",
             "screen_f",
         ]
+
+
+async def test_skip_action() -> None:
+    """Test that a binding may be skipped by an action returning False"""
+
+    class Handle(Widget, can_focus=True):
+        BINDINGS = [("t", "test('foo')", "Test")]
+
+        def action_test(self, text: str) -> None:
+            self.app.exit(text)
+
+    no_handle_invoked = False
+
+    class NoHandle(Widget, can_focus=True):
+        BINDINGS = [("t", "test('bar')", "Test")]
+
+        def action_test(self, text: str) -> bool:
+            nonlocal no_handle_invoked
+            no_handle_invoked = True
+            return False
+
+    class SkipApp(App):
+        def compose(self) -> ComposeResult:
+            yield Handle(NoHandle())
+
+        def on_mount(self) -> None:
+            self.query_one(NoHandle).focus()
+
+    async with SkipApp().run_test() as pilot:
+        # Check the NoHandle widget has focus
+        assert pilot.app.query_one(NoHandle).has_focus
+        # Press the "t" key
+        await pilot.press("t")
+        # Check the action on the no handle widget was called
+        assert no_handle_invoked
+        # Check the return value, confirming that the action on Handle was called
+        assert pilot.app.return_value == "foo"