diff --git a/requirements-test.txt b/requirements-test.txt
--- a/requirements-test.txt
+++ b/requirements-test.txt
@@ -1,4 +1,4 @@
-hypothesis >= 3.11.0, < 6
+hypothesis >= 3.11.0
 pytest
 pytest-mock
 sqlalchemy-stubs
diff --git a/swh/storage/tests/algos/test_snapshot.py b/swh/storage/tests/algos/test_snapshot.py
--- a/swh/storage/tests/algos/test_snapshot.py
+++ b/swh/storage/tests/algos/test_snapshot.py
@@ -3,7 +3,7 @@
 # License: GNU General Public License version 3, or any later version
 # See top-level LICENSE file for more information
 
-from hypothesis import given
+from hypothesis import given, settings
 import pytest
 
 from swh.model.hypothesis_strategies import branch_names, branch_targets, snapshots
@@ -21,6 +21,7 @@
     snapshot_resolve_alias,
     visits_and_snapshots_get_from_revision,
 )
+from swh.storage.tests.conftest import function_scoped_fixture_check
 from swh.storage.utils import now
 
 
@@ -32,6 +33,7 @@
     }
 
 
+@settings(suppress_health_check=function_scoped_fixture_check)
 @given(snapshot=snapshots(min_size=0, max_size=10, only_objects=False))
 def test_snapshot_small(swh_storage, snapshot):  # noqa
     swh_storage.snapshot_add([snapshot])
@@ -40,6 +42,7 @@
     assert snapshot == returned_snapshot
 
 
+@settings(suppress_health_check=function_scoped_fixture_check)
 @given(branch_name=branch_names(), branch_target=branch_targets(only_objects=True))
 def test_snapshot_large(swh_storage, branch_name, branch_target):  # noqa
     snapshot = Snapshot(
diff --git a/swh/storage/tests/conftest.py b/swh/storage/tests/conftest.py
--- a/swh/storage/tests/conftest.py
+++ b/swh/storage/tests/conftest.py
@@ -5,7 +5,7 @@
 
 import multiprocessing.util
 
-from hypothesis import settings
+from hypothesis import HealthCheck, settings
 import pytest
 
 try:
@@ -19,6 +19,13 @@
 from swh.model.tests.generate_testdata import gen_contents, gen_origins
 from swh.storage.interface import StorageInterface
 
+# we use getattr here to keep mypy happy regardless hypothesis version
+function_scoped_fixture_check = (
+    [getattr(HealthCheck, "function_scoped_fixture")]
+    if hasattr(HealthCheck, "function_scoped_fixture")
+    else []
+)
+
 # define tests profile. Full documentation is at:
 # https://hypothesis.readthedocs.io/en/latest/settings.html#settings-profiles
 settings.register_profile("fast", max_examples=5, deadline=5000)
diff --git a/swh/storage/tests/storage_tests.py b/swh/storage/tests/storage_tests.py
--- a/swh/storage/tests/storage_tests.py
+++ b/swh/storage/tests/storage_tests.py
@@ -36,6 +36,7 @@
 from swh.storage.common import origin_url_to_sha1 as sha1
 from swh.storage.exc import HashCollision, StorageArgumentException
 from swh.storage.interface import ListOrder, PagedResult, StorageInterface
+from swh.storage.tests.conftest import function_scoped_fixture_check
 from swh.storage.utils import (
     content_hex_hashes,
     now,
@@ -424,7 +425,10 @@
         assert missing == [cont2.hashes()]
 
     @pytest.mark.property_based
-    @settings(deadline=None)  # this test is very slow
+    @settings(
+        deadline=None,  # this test is very slow
+        suppress_health_check=function_scoped_fixture_check,
+    )
     @given(
         strategies.sets(
             elements=strategies.sampled_from(["sha256", "sha1_git", "blake2s256"]),
@@ -455,6 +459,7 @@
             ) == set(missing_per_hash[hash])
 
     @pytest.mark.property_based
+    @settings(suppress_health_check=function_scoped_fixture_check,)
     @given(
         strategies.sets(
             elements=strategies.sampled_from(["sha256", "sha1_git", "blake2s256"]),
@@ -3951,7 +3956,9 @@
         )
         assert swh_storage.origin_count("github", regexp=True, with_visit=True) == 1
 
-    @settings(suppress_health_check=[HealthCheck.too_slow])
+    @settings(
+        suppress_health_check=[HealthCheck.too_slow] + function_scoped_fixture_check,
+    )
     @given(strategies.lists(objects(split_content=True), max_size=2))
     def test_add_arbitrary(self, swh_storage, objects):
         for (obj_type, obj) in objects: