diff --git a/src/textual/_compositor.py b/src/textual/_compositor.py
index 9ccb73099f..f1aaf4681a 100644
--- a/src/textual/_compositor.py
+++ b/src/textual/_compositor.py
@@ -559,6 +559,8 @@ def _arrange_root(
         map: CompositorMap = {}
         widgets: set[Widget] = set()
         add_new_widget = widgets.add
+        invisible_widgets: set[Widget] = set()
+        add_new_invisible_widget = invisible_widgets.add
         layer_order: int = 0
 
         no_clip = size.region
@@ -595,6 +597,8 @@ def add_widget(
 
             if visible:
                 add_new_widget(widget)
+            else:
+                add_new_invisible_widget(widget)
             styles_offset = styles.offset
             layout_offset = (
                 styles_offset.resolve(region.size, clip.size)
@@ -740,6 +744,7 @@ def add_widget(
             True,
             NULL_SPACING,
         )
+        widgets -= invisible_widgets
         return map, widgets
 
     @property
diff --git a/tests/test_widget_visibility.py b/tests/test_widget_visibility.py
new file mode 100644
index 0000000000..c625771ff0
--- /dev/null
+++ b/tests/test_widget_visibility.py
@@ -0,0 +1,34 @@
+from textual.app import App, ComposeResult
+from textual.widgets import Label
+
+
+async def test_hide() -> None:
+    """Check that setting visibility produces Hide messages."""
+    # https://github.com/Textualize/textual/issues/3460
+    visibility: list[bool] = []
+
+    class ShowHideLabel(Label):
+        def on_show(self) -> None:
+            visibility.append(True)
+
+        def on_hide(self) -> None:
+            visibility.append(False)
+
+    class ShowHideApp(App[None]):
+        BINDINGS = [("space", "toggle_label")]
+
+        def compose(self) -> ComposeResult:
+            yield ShowHideLabel("Here I am")
+
+        def action_toggle_label(self) -> None:
+            self.query_one(ShowHideLabel).visible = not self.query_one(
+                ShowHideLabel
+            ).visible
+
+    app = ShowHideApp()
+    async with app.run_test() as pilot:
+        assert visibility == [True]
+        await pilot.press("space")
+        assert visibility == [True, False]
+        await pilot.press("space")
+        assert visibility == [True, False, True]