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 ae1b9049b3a..ced0bc8003e 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.ClearScrollResponse;
 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;
@@ -129,6 +130,7 @@ import org.elasticsearch.indices.IndicesQueryCache;
 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;
@@ -2392,4 +2394,22 @@ public abstract class ESIntegTestCase extends ESTestCase {
     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/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 5619c3b6890..c735a91395c 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 class BasicDistributedJobsIT extends BaseMlIntegTestCase {
     }
 
     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 dc0af77dc8a..5b1c1b52051 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
@@ -373,6 +373,9 @@ public class MlDistributedFailureIT extends BaseMlIntegTestCase {
 
     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 569aee4c261..044cb712f49 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 class TooManyJobsIT extends BaseMlIntegTestCase {
     }
 
     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());
     }