diff --git a/src/lightning/fabric/loggers/tensorboard.py b/src/lightning/fabric/loggers/tensorboard.py
index 208244dc38cd3..010e8b55636c3 100644
--- a/src/lightning/fabric/loggers/tensorboard.py
+++ b/src/lightning/fabric/loggers/tensorboard.py
@@ -211,8 +211,11 @@ def log_metrics(self, metrics: Mapping[str, float], step: Optional[int] = None)
             else:
                 try:
                     self.experiment.add_scalar(k, v, step)
-                # TODO(fabric): specify the possible exception
-                except Exception as ex:
+                except (
+                    NotImplementedError,
+                    ValueError,
+                    ModuleNotFoundError,  # https://github.com/pytorch/pytorch/issues/24175
+                ) as ex:
                     raise ValueError(
                         f"\n you tried to log {v} which is currently not supported. Try a dict or a scalar/tensor."
                     ) from ex
diff --git a/tests/tests_pytorch/loggers/test_tensorboard.py b/tests/tests_pytorch/loggers/test_tensorboard.py
index 7e02a73c93082..751224b93d3a6 100644
--- a/tests/tests_pytorch/loggers/test_tensorboard.py
+++ b/tests/tests_pytorch/loggers/test_tensorboard.py
@@ -12,6 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 import os
+import re
 from argparse import Namespace
 from unittest import mock
 from unittest.mock import Mock
@@ -157,6 +158,18 @@ def test_tensorboard_log_metrics(tmp_path, step_idx):
     logger.log_metrics(metrics, step_idx)
 
 
+@pytest.mark.parametrize("value", [[1], "x", None])
+def test_tensorboard_log_metrics_exception_message(tmp_path, value):
+    logger = TensorBoardLogger(tmp_path)
+    with pytest.raises(
+        ValueError,
+        match=re.escape(
+            f"you tried to log {value} which is currently not supported. Try a dict or a scalar/tensor.",
+        ),
+    ):
+        logger.log_metrics(metrics={"metric": value})
+
+
 def test_tensorboard_log_hyperparams(tmp_path):
     logger = TensorBoardLogger(tmp_path)
     hparams = {