diff --git a/src/humanize/time.py b/src/humanize/time.py
index 4a07d52..18a58d5 100644
--- a/src/humanize/time.py
+++ b/src/humanize/time.py
@@ -148,11 +148,16 @@ def naturaldelta(
         delta = value
     else:
         try:
-            int(value)  # Explicitly don't support string such as "NaN" or "inf"
-            value = float(value)
-            delta = dt.timedelta(seconds=value)
-        except (ValueError, TypeError):
+            # Explicitly don't support non-finite or non-numeric values such as
+            # "NaN"/"inf": int() raises ValueError for NaN and OverflowError for
+            # infinity, and TypeError for unsupported types.
+            int(value)
+        except (ValueError, TypeError, OverflowError):
             return str(value)
+        value = float(value)
+        # A finite value that is too large for timedelta still raises
+        # OverflowError here, as documented.
+        delta = dt.timedelta(seconds=value)
 
     use_months = months
 
diff --git a/tests/test_time.py b/tests/test_time.py
index 7699770..1d757ba 100644
--- a/tests/test_time.py
+++ b/tests/test_time.py
@@ -138,6 +138,27 @@ def test_naturaldelta(test_input: float | dt.timedelta, expected: str) -> None:
         assert humanize.naturaldelta(-test_input) == expected
 
 
+@pytest.mark.parametrize(
+    "test_input, expected",
+    [
+        (float("nan"), "nan"),
+        (float("inf"), "inf"),
+        (float("-inf"), "-inf"),
+    ],
+)
+def test_naturaldelta_nonfinite(test_input: float, expected: str) -> None:
+    # Non-finite floats are returned unchanged, per the docstring. `nan` raises
+    # ValueError from int() while `inf` raises OverflowError; both are caught.
+    assert humanize.naturaldelta(test_input) == expected
+
+
+def test_naturaldelta_overflow() -> None:
+    # A finite value too large to convert to a timedelta still raises, as
+    # documented, and is not swallowed by the non-finite guard.
+    with pytest.raises(OverflowError):
+        humanize.naturaldelta(1e40)
+
+
 @freeze_time(FROZEN_DATE)
 @pytest.mark.parametrize(
     "test_input, expected",