diff --git a/src/_zkapauthorizer/tests/test_storage_protocol.py b/src/_zkapauthorizer/tests/test_storage_protocol.py
index 98320fdd1581c2a537bc505492c597ddceb0c3e8..b0b105df47a10151091e998c109d544925288c4d 100644
--- a/src/_zkapauthorizer/tests/test_storage_protocol.py
+++ b/src/_zkapauthorizer/tests/test_storage_protocol.py
@@ -140,6 +140,22 @@ class LocalRemote(object):
         )
 
 
+def assume_one_pass(test_and_write_vectors_for_shares):
+    """
+    Assume that the writes represented by the given ``TestAndWriteVectors``
+    will cost at most one pass.
+    """
+    from .._storage_server import (
+        BYTES_PER_PASS,
+        get_sharenums,
+        get_allocated_size,
+        required_passes,
+    )
+    tw_vectors = {k: v.for_call() for (k, v) in test_and_write_vectors_for_shares.items()}
+    sharenums = get_sharenums(tw_vectors)
+    allocated_size = get_allocated_size(tw_vectors)
+    assume(required_passes(BYTES_PER_PASS, sharenums, allocated_size) <= 1)
+
 
 class ShareTests(TestCase):
     """
@@ -194,6 +210,9 @@ class ShareTests(TestCase):
         resulting buckets can be read back using *get_buckets* and methods of
         those resulting buckets.
         """
+        # XXX
+        assume(len(sharenums) * size < 128 * 1024 * 10)
+
         # Hypothesis causes our storage server to be used many times.  Clean
         # up between iterations.
         cleanup_storage_server(self.anonymous_storage_server)
@@ -381,6 +400,9 @@ class ShareTests(TestCase):
         Mutable share data written using *slot_testv_and_readv_and_writev* can be
         read back.
         """
+        # XXX
+        assume_one_pass(test_and_write_vectors_for_shares)
+
         # Hypothesis causes our storage server to be used many times.  Clean
         # up between iterations.
         cleanup_storage_server(self.anonymous_storage_server)
@@ -458,6 +480,9 @@ class ShareTests(TestCase):
         When mutable share data is written using *slot_testv_and_readv_and_writev*
         any leases on the corresponding slot remain the same.
         """
+        # XXX
+        assume_one_pass(test_and_write_vectors_for_shares)
+
         # Hypothesis causes our storage server to be used many times.  Clean
         # up between iterations.
         cleanup_storage_server(self.anonymous_storage_server)