diff --git a/test/e2e/framework/pv_util.go b/test/e2e/framework/pv_util.go
index 619cbab0f7e..91945ffda6f 100644
--- a/test/e2e/framework/pv_util.go
+++ b/test/e2e/framework/pv_util.go
@@ -950,6 +950,36 @@ func SetAntiAffinity(nodeSelection *NodeSelection, nodeName string) {
 	SetNodeAffinityRequirement(nodeSelection, v1.NodeSelectorOpNotIn, nodeName)
 }
 
+// SetNodeAffinityPreference sets affinity preference with specified operator to nodeName to nodeSelection
+func SetNodeAffinityPreference(nodeSelection *NodeSelection, operator v1.NodeSelectorOperator, nodeName string) {
+	// Add node-anti-affinity.
+	if nodeSelection.Affinity == nil {
+		nodeSelection.Affinity = &v1.Affinity{}
+	}
+	if nodeSelection.Affinity.NodeAffinity == nil {
+		nodeSelection.Affinity.NodeAffinity = &v1.NodeAffinity{}
+	}
+	nodeSelection.Affinity.NodeAffinity.PreferredDuringSchedulingIgnoredDuringExecution = append(nodeSelection.Affinity.NodeAffinity.PreferredDuringSchedulingIgnoredDuringExecution,
+		v1.PreferredSchedulingTerm{
+			Weight: int32(100),
+			Preference: v1.NodeSelectorTerm{
+				MatchFields: []v1.NodeSelectorRequirement{
+					{Key: "metadata.name", Operator: operator, Values: []string{nodeName}},
+				},
+			},
+		})
+}
+
+// SetAffinityPreference sets affinity preference to nodeName to nodeSelection
+func SetAffinityPreference(nodeSelection *NodeSelection, nodeName string) {
+	SetNodeAffinityPreference(nodeSelection, v1.NodeSelectorOpIn, nodeName)
+}
+
+// SetAntiAffinityPreference sets anti-affinity preference to nodeName to nodeSelection
+func SetAntiAffinityPreference(nodeSelection *NodeSelection, nodeName string) {
+	SetNodeAffinityPreference(nodeSelection, v1.NodeSelectorOpNotIn, nodeName)
+}
+
 // CreateClientPod defines and creates a pod with a mounted PV.  Pod runs infinite loop until killed.
 func CreateClientPod(c clientset.Interface, ns string, pvc *v1.PersistentVolumeClaim) (*v1.Pod, error) {
 	return CreatePod(c, ns, nil, []*v1.PersistentVolumeClaim{pvc}, true, "")
diff --git a/test/e2e/storage/testsuites/multivolume.go b/test/e2e/storage/testsuites/multivolume.go
index ac3a08a96d9..c896fd5d9d1 100644
--- a/test/e2e/storage/testsuites/multivolume.go
+++ b/test/e2e/storage/testsuites/multivolume.go
@@ -375,7 +375,9 @@ func TestAccessMultipleVolumesAcrossPodRecreation(f *framework.Framework, cs cli
 	if requiresSameNode {
 		framework.SetAffinity(&node, nodeName)
 	} else {
-		framework.SetAntiAffinity(&node, nodeName)
+		// Do not put hard requirement on the anti-affinity. In some occasions there might exist only
+		// one available node for the pod scheduling (e.g., multi-AZ cluster with one node per AZ)
+		framework.SetAntiAffinityPreference(&node, nodeName)
 	}
 
 	// Test access to multiple volumes again on the node updated above
@@ -383,7 +385,13 @@ func TestAccessMultipleVolumesAcrossPodRecreation(f *framework.Framework, cs cli
 	readSeedBase = writeSeedBase
 	// Update writeSeed with new value
 	writeSeedBase = time.Now().UTC().UnixNano()
-	_ = testAccessMultipleVolumes(f, cs, ns, node, pvcs, readSeedBase, writeSeedBase)
+	secondNodeName := testAccessMultipleVolumes(f, cs, ns, node, pvcs, readSeedBase, writeSeedBase)
+	if !requiresSameNode && (secondNodeName == nodeName) {
+		// The pod was created on the same node: presumably there was no other node available
+		// for the second pod scheduling -- this does not mean the test should fail. Skip it instead/
+		e2elog.Logf("Warning: The pod got scheduled on the same node despite requesting otherwise: skipping test")
+		framework.Skipf("No node available for the second pod found")
+	}
 }
 
 // TestConcurrentAccessToSingleVolume tests access to a single volume from multiple pods,
@@ -395,6 +403,7 @@ func TestConcurrentAccessToSingleVolume(f *framework.Framework, cs clientset.Int
 
 	var pods []*v1.Pod
 
+	firstNodeName := ""
 	// Create each pod with pvc
 	for i := 0; i < numPods; i++ {
 		index := i + 1
@@ -411,12 +420,22 @@ func TestConcurrentAccessToSingleVolume(f *framework.Framework, cs clientset.Int
 		pods = append(pods, pod)
 		framework.ExpectNoError(err, fmt.Sprintf("get pod%d", index))
 		actualNodeName := pod.Spec.NodeName
+		// Remember where the first pod was scheduled
+		if i == 0 {
+			firstNodeName = actualNodeName
+		}
+		// If the second pod got scheduled on the same node as the first one it means
+		// there was no available node for the second pod and the test should be skipped.
+		if !requiresSameNode && i == 1 && (actualNodeName == firstNodeName) {
+			e2elog.Logf("Warning: The pod got scheduled on the same node as the previous one: skipping test")
+			framework.Skipf("No node available for the second pod found")
+		}
 
 		// Set affinity depending on requiresSameNode
 		if requiresSameNode {
 			framework.SetAffinity(&node, actualNodeName)
 		} else {
-			framework.SetAntiAffinity(&node, actualNodeName)
+			framework.SetAntiAffinityPreference(&node, actualNodeName)
 		}
 	}
 
diff --git a/test/e2e/storage/testsuites/provisioning.go b/test/e2e/storage/testsuites/provisioning.go
index d302666785a..8fc3a806e01 100644
--- a/test/e2e/storage/testsuites/provisioning.go
+++ b/test/e2e/storage/testsuites/provisioning.go
@@ -424,7 +424,9 @@ func PVMultiNodeCheck(client clientset.Interface, claim *v1.PersistentVolumeClai
 
 	// Add node-anti-affinity.
 	secondNode := node
-	framework.SetAntiAffinity(&secondNode, actualNodeName)
+	// Set anti-affinity preference: in case there are no nodes in the same AZ it may happen the second pod gets
+	// scheduled on the same node as the first one. In such a case the test needs to be skipped.
+	framework.SetAntiAffinityPreference(&secondNode, actualNodeName)
 	ginkgo.By(fmt.Sprintf("checking the created volume is readable and retains data on another node %+v", secondNode))
 	command = "grep 'hello world' /mnt/test/data"
 	if framework.NodeOSDistroIs("windows") {
@@ -434,9 +436,13 @@ func PVMultiNodeCheck(client clientset.Interface, claim *v1.PersistentVolumeClai
 	framework.ExpectNoError(e2epod.WaitForPodSuccessInNamespaceSlow(client, pod.Name, pod.Namespace))
 	runningPod, err = client.CoreV1().Pods(pod.Namespace).Get(pod.Name, metav1.GetOptions{})
 	framework.ExpectNoError(err, "get pod")
-	gomega.Expect(runningPod.Spec.NodeName).NotTo(gomega.Equal(actualNodeName), "second pod should have run on a different node")
 	StopPod(client, pod)
 	pod = nil
+	// The second pod got scheduled on the same node as the first one: skip the test.
+	if runningPod.Spec.NodeName == actualNodeName {
+		e2elog.Logf("Warning: The reader pod got scheduled on the same node as the writer pod: skipping test")
+		framework.Skipf("No node available for the second pod found")
+	}
 }
 
 // TestBindingWaitForFirstConsumer tests the binding with WaitForFirstConsumer mode