diff --git a/test/framework/src/main/java/org/elasticsearch/test/ESIntegTestCase.java b/test/framework/src/main/java/org/elasticsearch/test/ESIntegTestCase.java
index 2d3a5c28a76fc..9abd7fd064827 100644
--- a/test/framework/src/main/java/org/elasticsearch/test/ESIntegTestCase.java
+++ b/test/framework/src/main/java/org/elasticsearch/test/ESIntegTestCase.java
@@ -57,6 +57,7 @@
 import org.elasticsearch.action.search.SearchResponse;
 import org.elasticsearch.action.support.DefaultShardOperationFailedException;
 import org.elasticsearch.action.support.IndicesOptions;
+import org.elasticsearch.bootstrap.JavaVersion;
 import org.elasticsearch.client.AdminClient;
 import org.elasticsearch.client.Client;
 import org.elasticsearch.client.ClusterAdminClient;
@@ -132,6 +133,7 @@
 import org.elasticsearch.indices.IndicesRequestCache;
 import org.elasticsearch.indices.store.IndicesStore;
 import org.elasticsearch.ingest.IngestMetadata;
+import org.elasticsearch.monitor.os.OsInfo;
 import org.elasticsearch.node.NodeMocksPlugin;
 import org.elasticsearch.plugins.NetworkPlugin;
 import org.elasticsearch.plugins.Plugin;
@@ -2411,4 +2413,22 @@ public static String resolveCustomDataPath(String index) {
     public static boolean inFipsJvm() {
         return Boolean.parseBoolean(System.getProperty(FIPS_SYSPROP));
     }
+
+    /**
+     * On Debian 8 the "memory" subsystem is not mounted by default
+     * when cgroups are enabled, and this confuses many versions of
+     * Java prior to Java 15.  Tests that rely on machine memory
+     * being accurately determined will not work on such setups,
+     * and can use this method for selective muting.
+     * See https://github.com/elastic/elasticsearch/issues/67089
+     * and https://github.com/elastic/elasticsearch/issues/66885
+     */
+    protected boolean willSufferDebian8MemoryProblem() {
+        final NodesInfoResponse response = client().admin().cluster().prepareNodesInfo().execute().actionGet();
+        final boolean anyDebian8Nodes = response.getNodes()
+            .stream()
+            .anyMatch(ni -> ni.getInfo(OsInfo.class).getPrettyName().equals("Debian GNU/Linux 8 (jessie)"));
+        boolean java15Plus = JavaVersion.current().compareTo(JavaVersion.parse("15")) >= 0;
+        return anyDebian8Nodes && java15Plus == false;
+    }
 }
diff --git a/x-pack/plugin/autoscaling/src/internalClusterTest/java/org/elasticsearch/xpack/autoscaling/action/TransportGetAutoscalingCapacityActionIT.java b/x-pack/plugin/autoscaling/src/internalClusterTest/java/org/elasticsearch/xpack/autoscaling/action/TransportGetAutoscalingCapacityActionIT.java
index 5a4dd40f2f08b..ad7de56752181 100644
--- a/x-pack/plugin/autoscaling/src/internalClusterTest/java/org/elasticsearch/xpack/autoscaling/action/TransportGetAutoscalingCapacityActionIT.java
+++ b/x-pack/plugin/autoscaling/src/internalClusterTest/java/org/elasticsearch/xpack/autoscaling/action/TransportGetAutoscalingCapacityActionIT.java
@@ -6,10 +6,7 @@
 
 package org.elasticsearch.xpack.autoscaling.action;
 
-import org.elasticsearch.action.admin.cluster.node.info.NodesInfoResponse;
-import org.elasticsearch.bootstrap.JavaVersion;
 import org.elasticsearch.env.NodeEnvironment;
-import org.elasticsearch.monitor.os.OsInfo;
 import org.elasticsearch.monitor.os.OsProbe;
 import org.elasticsearch.test.ESIntegTestCase;
 import org.elasticsearch.xpack.autoscaling.AutoscalingIntegTestCase;
@@ -26,13 +23,8 @@
 public class TransportGetAutoscalingCapacityActionIT extends AutoscalingIntegTestCase {
 
     public void testCurrentCapacity() throws Exception {
-        final NodesInfoResponse response = client().admin().cluster().prepareNodesInfo().execute().actionGet();
-        final boolean anyDebian8Nodes = response.getNodes()
-            .stream()
-            .anyMatch(ni -> ni.getInfo(OsInfo.class).getPrettyName().equals("Debian GNU/Linux 8 (jessie)"));
-        boolean java15Plus = JavaVersion.current().compareTo(JavaVersion.parse("15")) >= 0;
         // see: https://github.com/elastic/elasticsearch/issues/67089#issuecomment-756114654
-        assumeTrue("cannot run on debian 8 prior to java 15", java15Plus || anyDebian8Nodes == false);
+        assumeFalse("cannot run on debian 8 prior to java 15", willSufferDebian8MemoryProblem());
 
         assertThat(capacity().results().keySet(), Matchers.empty());
         long memory = OsProbe.getInstance().getTotalPhysicalMemorySize();
diff --git a/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/BasicDistributedJobsIT.java b/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/BasicDistributedJobsIT.java
index eab3ade5c202a..03bb86f417bf2 100644
--- a/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/BasicDistributedJobsIT.java
+++ b/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/BasicDistributedJobsIT.java
@@ -423,6 +423,10 @@ public void testMlStateAndResultsIndicesNotAvailable() throws Exception {
     }
 
     public void testCloseUnassignedLazyJobAndDatafeed() throws Exception {
+
+        // see: https://github.com/elastic/elasticsearch/issues/66885#issuecomment-758790179
+        assumeFalse("cannot run on debian 8 prior to java 15", willSufferDebian8MemoryProblem());
+
         internalCluster().ensureAtLeastNumDataNodes(3);
         ensureStableCluster(3);
 
diff --git a/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java b/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java
index fa98bd74b427c..1ac229027f01b 100644
--- a/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java
+++ b/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java
@@ -408,6 +408,9 @@ public void testStopAndForceStopDatafeed() throws Exception {
 
     public void testJobRelocationIsMemoryAware() throws Exception {
 
+        // see: https://github.com/elastic/elasticsearch/issues/66885#issuecomment-758790179
+        assumeFalse("cannot run on debian 8 prior to java 15", willSufferDebian8MemoryProblem());
+
         internalCluster().ensureAtLeastNumDataNodes(1);
         ensureStableCluster();
 
diff --git a/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java b/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java
index 7e1f4c845f90a..14e499117dbc1 100644
--- a/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java
+++ b/x-pack/plugin/ml/src/internalClusterTest/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java
@@ -126,10 +126,14 @@ public void testLazyNodeValidation() throws Exception {
     }
 
     public void testSingleNode() throws Exception {
+        // see: https://github.com/elastic/elasticsearch/issues/66885#issuecomment-758790179
+        assumeFalse("cannot run on debian 8 prior to java 15", willSufferDebian8MemoryProblem());
         verifyMaxNumberOfJobsLimit(1, randomIntBetween(1, 20), randomBoolean());
     }
 
     public void testMultipleNodes() throws Exception {
+        // see: https://github.com/elastic/elasticsearch/issues/66885#issuecomment-758790179
+        assumeFalse("cannot run on debian 8 prior to java 15", willSufferDebian8MemoryProblem());
         verifyMaxNumberOfJobsLimit(3, randomIntBetween(1, 20), randomBoolean());
     }