diff --git a/t/t5500-fetch-pack.sh b/t/t5500-fetch-pack.sh
index 49c540b1e1..0ef4d6f20c 100755
--- a/t/t5500-fetch-pack.sh
+++ b/t/t5500-fetch-pack.sh
@@ -636,7 +636,9 @@ test_expect_success 'fetch-pack cannot fetch a raw sha1 that is not advertised a
 	test_commit -C server 6 &&
 
 	git init client &&
-	test_must_fail git -C client fetch-pack ../server \
+	# Some protocol versions (e.g. 2) support fetching
+	# unadvertised objects, so restrict this test to v0.
+	test_must_fail env GIT_TEST_PROTOCOL_VERSION= git -C client fetch-pack ../server \
 		$(git -C server rev-parse refs/heads/master^) 2>err &&
 	test_i18ngrep "Server does not allow request for unadvertised object" err
 '
diff --git a/t/t5516-fetch-push.sh b/t/t5516-fetch-push.sh
index 37e8e80893..4bfbb79654 100755
--- a/t/t5516-fetch-push.sh
+++ b/t/t5516-fetch-push.sh
@@ -1147,8 +1147,12 @@ test_expect_success 'fetch exact SHA1' '
 		git prune &&
 		test_must_fail git cat-file -t $the_commit &&
 
+		# Some protocol versions (e.g. 2) support fetching
+		# unadvertised objects, so restrict this test to v0.
+
 		# fetching the hidden object should fail by default
-		test_must_fail git fetch -v ../testrepo $the_commit:refs/heads/copy 2>err &&
+		test_must_fail env GIT_TEST_PROTOCOL_VERSION= \
+			git fetch -v ../testrepo $the_commit:refs/heads/copy 2>err &&
 		test_i18ngrep "Server does not allow request for unadvertised object" err &&
 		test_must_fail git rev-parse --verify refs/heads/copy &&
 
@@ -1204,7 +1208,10 @@ do
 		mk_empty shallow &&
 		(
 			cd shallow &&
-			test_must_fail git fetch --depth=1 ../testrepo/.git $SHA1 &&
+			# Some protocol versions (e.g. 2) support fetching
+			# unadvertised objects, so restrict this test to v0.
+			test_must_fail env GIT_TEST_PROTOCOL_VERSION= \
+				git fetch --depth=1 ../testrepo/.git $SHA1 &&
 			git --git-dir=../testrepo/.git config uploadpack.allowreachablesha1inwant true &&
 			git fetch --depth=1 ../testrepo/.git $SHA1 &&
 			git cat-file commit $SHA1
@@ -1232,15 +1239,20 @@ do
 		mk_empty shallow &&
 		(
 			cd shallow &&
-			test_must_fail ok=sigpipe git fetch ../testrepo/.git $SHA1_3 &&
-			test_must_fail ok=sigpipe git fetch ../testrepo/.git $SHA1_1 &&
+			# Some protocol versions (e.g. 2) support fetching
+			# unadvertised objects, so restrict this test to v0.
+			test_must_fail ok=sigpipe env GIT_TEST_PROTOCOL_VERSION= \
+				git fetch ../testrepo/.git $SHA1_3 &&
+			test_must_fail ok=sigpipe env GIT_TEST_PROTOCOL_VERSION= \
+				git fetch ../testrepo/.git $SHA1_1 &&
 			git --git-dir=../testrepo/.git config uploadpack.allowreachablesha1inwant true &&
 			git fetch ../testrepo/.git $SHA1_1 &&
 			git cat-file commit $SHA1_1 &&
 			test_must_fail git cat-file commit $SHA1_2 &&
 			git fetch ../testrepo/.git $SHA1_2 &&
 			git cat-file commit $SHA1_2 &&
-			test_must_fail ok=sigpipe git fetch ../testrepo/.git $SHA1_3
+			test_must_fail ok=sigpipe env GIT_TEST_PROTOCOL_VERSION= \
+				git fetch ../testrepo/.git $SHA1_3
 		)
 	'
 done
diff --git a/t/t5551-http-fetch-smart.sh b/t/t5551-http-fetch-smart.sh
index a852684e16..f02ae3b797 100755
--- a/t/t5551-http-fetch-smart.sh
+++ b/t/t5551-http-fetch-smart.sh
@@ -307,7 +307,10 @@ test_expect_success 'test allowreachablesha1inwant with unreachable' '
 
 	git init --bare test_reachable.git &&
 	git -C test_reachable.git remote add origin "$HTTPD_URL/smart/repo.git" &&
-	test_must_fail git -C test_reachable.git fetch origin "$(git rev-parse HEAD)"
+	# Some protocol versions (e.g. 2) support fetching
+	# unadvertised objects, so restrict this test to v0.
+	test_must_fail env GIT_TEST_PROTOCOL_VERSION= \
+		git -C test_reachable.git fetch origin "$(git rev-parse HEAD)"
 '
 
 test_expect_success 'test allowanysha1inwant with unreachable' '
@@ -326,7 +329,10 @@ test_expect_success 'test allowanysha1inwant with unreachable' '
 
 	git init --bare test_reachable.git &&
 	git -C test_reachable.git remote add origin "$HTTPD_URL/smart/repo.git" &&
-	test_must_fail git -C test_reachable.git fetch origin "$(git rev-parse HEAD)" &&
+	# Some protocol versions (e.g. 2) support fetching
+	# unadvertised objects, so restrict this test to v0.
+	test_must_fail env GIT_TEST_PROTOCOL_VERSION= \
+		git -C test_reachable.git fetch origin "$(git rev-parse HEAD)" &&
 
 	git -C "$server" config uploadpack.allowanysha1inwant 1 &&
 	git -C test_reachable.git fetch origin "$(git rev-parse HEAD)"
diff --git a/t/t7406-submodule-update.sh b/t/t7406-submodule-update.sh
index e87164aa8f..c973278300 100755
--- a/t/t7406-submodule-update.sh
+++ b/t/t7406-submodule-update.sh
@@ -943,7 +943,10 @@ test_expect_success 'submodule update clone shallow submodule outside of depth'
 		cd super3 &&
 		sed -e "s#url = ../#url = file://$pwd/#" <.gitmodules >.gitmodules.tmp &&
 		mv -f .gitmodules.tmp .gitmodules &&
-		test_must_fail git submodule update --init --depth=1 2>actual &&
+		# Some protocol versions (e.g. 2) support fetching
+		# unadvertised objects, so restrict this test to v0.
+		test_must_fail env GIT_TEST_PROTOCOL_VERSION= \
+			git submodule update --init --depth=1 2>actual &&
 		test_i18ngrep "Direct fetching of that commit failed." actual &&
 		git -C ../submodule config uploadpack.allowReachableSHA1InWant true &&
 		git submodule update --init --depth=1 >actual &&
