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 04477e2c117ba..7759618347e3f 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.Requests;
@@ -2350,4 +2351,22 @@ public static Index resolveIndex(String index) {
     public static boolean inFipsJvm() {
         return Security.getProviders()[0].getName().toLowerCase(Locale.ROOT).contains("fips");
     }
+
+    /**
+     * 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.getOs().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/test/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java b/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java
index 940b219aac881..2b2a1b9cb7f06 100644
--- a/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java
+++ b/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/integration/MlDistributedFailureIT.java
@@ -337,6 +337,9 @@ 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);
         ensureStableClusterOnAllNodes(1);
 
diff --git a/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java b/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java
index f79e9f1e4e945..0beae14be954f 100644
--- a/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java
+++ b/x-pack/plugin/ml/src/test/java/org/elasticsearch/xpack/ml/integration/TooManyJobsIT.java
@@ -124,10 +124,14 @@ 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));
     }
 
     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));
     }