diff --git a/loky/backend/context.py b/loky/backend/context.py
index 62795d65..d8cc59ba 100644
--- a/loky/backend/context.py
+++ b/loky/backend/context.py
@@ -145,22 +145,34 @@ def _cpu_count_cgroup(os_cpu_count):
     cpu_max_fname = "/sys/fs/cgroup/cpu.max"
     cfs_quota_fname = "/sys/fs/cgroup/cpu/cpu.cfs_quota_us"
     cfs_period_fname = "/sys/fs/cgroup/cpu/cpu.cfs_period_us"
+
+    cpu_quota_us = None
+    cpu_period_us = None
+
     if os.path.exists(cpu_max_fname):
         # cgroup v2
         # https://www.kernel.org/doc/html/latest/admin-guide/cgroup-v2.html
         with open(cpu_max_fname) as fh:
-            cpu_quota_us, cpu_period_us = fh.read().strip().split()
-    elif os.path.exists(cfs_quota_fname) and os.path.exists(cfs_period_fname):
-        # cgroup v1
-        # https://www.kernel.org/doc/html/latest/scheduler/sched-bwc.html#management
-        with open(cfs_quota_fname) as fh:
-            cpu_quota_us = fh.read().strip()
-        with open(cfs_period_fname) as fh:
-            cpu_period_us = fh.read().strip()
-    else:
-        # No Cgroup CPU bandwidth limit (e.g. non-Linux platform)
-        cpu_quota_us = "max"
-        cpu_period_us = 100_000  # unused, for consistency with default values
+            # Parse the quota and period values
+            parts = fh.read().strip().split()
+            if len(parts) == 2:
+                cpu_quota_us, cpu_period_us = parts
+            # If len(parts) != 2, leave as None and fall back to v1
+
+    # If we didn't get values from cgroup v2, try cgroup v1
+    if cpu_quota_us is None or cpu_period_us is None:
+        if os.path.exists(cfs_quota_fname) and os.path.exists(
+            cfs_period_fname
+        ):
+            # cgroup v1
+            # https://www.kernel.org/doc/html/latest/scheduler/sched-bwc.html#management
+            with open(cfs_quota_fname) as fh:
+                cpu_quota_us = fh.read().strip()
+            with open(cfs_period_fname) as fh:
+                cpu_period_us = fh.read().strip()
+        else:
+            # No Cgroup CPU bandwidth limit (e.g. non-Linux platform)
+            cpu_quota_us = "max"
 
     if cpu_quota_us == "max":
         # No active Cgroup quota on a Cgroup-capable platform
@@ -172,7 +184,7 @@ def _cpu_count_cgroup(os_cpu_count):
             return math.ceil(cpu_quota_us / cpu_period_us)
         else:  # pragma: no cover
             # Setting a negative cpu_quota_us value is a valid way to disable
-            # cgroup CPU bandwith limits
+            # cgroup CPU bandwidth limits
             return os_cpu_count
 
 
diff --git a/tests/test_loky_module.py b/tests/test_loky_module.py
index 3fd1b542..ef1194c3 100644
--- a/tests/test_loky_module.py
+++ b/tests/test_loky_module.py
@@ -6,6 +6,7 @@
 import tempfile
 import warnings
 from subprocess import check_output
+from unittest.mock import patch, mock_open
 
 import pytest
 
@@ -142,7 +143,27 @@ def test_cpu_count_cgroup_limit():
         os.path.join(loky_module_path, os.pardir)
     )
 
-    # The following will always run using the Python 3.7 docker image.
+    # Check if Docker can actually set cgroup CPU limits in this environment
+    # by verifying that --cpus flag writes to cgroup files
+    cgroup_check = check_output(
+        f'{docker_bin} run --rm --cpus 0.5 python:3.10 python3 -c "'
+        "import os; "
+        "v2 = '/sys/fs/cgroup/cpu.max'; "
+        "v1_quota = '/sys/fs/cgroup/cpu/cpu.cfs_quota_us'; "
+        "v2_content = open(v2).read().strip() if os.path.exists(v2) else ''; "
+        "v1_content = open(v1_quota).read().strip() if os.path.exists(v1_quota) else ''; "
+        "print('ok' if (v2_content and v2_content != 'max') or (v1_content and v1_content != '-1') else 'skip')"
+        '"',
+        shell=True,
+        text=True,
+    ).strip()
+
+    if cgroup_check != "ok":
+        pytest.skip(
+            "Docker doesn't properly set cgroup CPU limits in this environment"
+        )
+
+    # The following will always run using the Python 3.10 docker image.
     # We mount the loky source as /loky inside the container,
     # so it can be imported when running commands under /
 
@@ -238,3 +259,35 @@ def test_only_physical_cores_with_user_limitation():
     if cpu_count_user < cpu_count_mp:
         assert cpu_count() == cpu_count_user
         assert cpu_count(only_physical_cores=True) == cpu_count_user
+
+
+@pytest.mark.parametrize(
+    "read_data,description",
+    [
+        ("", "empty file"),
+        ("max\n", "max value"),
+    ],
+)
+def test_cpu_count_cgroup_invalid_content(read_data, description):
+    # Test that invalid cgroup cpu.max file content is handled gracefully
+    # and doesn't cause a ValueError when trying to unpack values
+    if sys.platform != "linux":
+        pytest.skip()
+
+    from loky.backend.context import _cpu_count_cgroup
+
+    os_cpu_count = mp.cpu_count()
+
+    # Mock the file with the provided read_data
+    with patch("builtins.open", mock_open(read_data=read_data)):
+        with patch("os.path.exists") as mock_exists:
+            # cpu.max exists, but other files don't
+            mock_exists.side_effect = (
+                lambda path: path == "/sys/fs/cgroup/cpu.max"
+            )
+
+            # This should not raise ValueError and return os_cpu_count
+            result = _cpu_count_cgroup(os_cpu_count)
+            assert (
+                result == os_cpu_count
+            ), f"cpu.max with {description} should return os_cpu_count"