diff --git a/client/rest/src/test/java/org/elasticsearch/client/RestClientBuilderIntegTests.java b/client/rest/src/test/java/org/elasticsearch/client/RestClientBuilderIntegTests.java
index 30359ea90f666..0d1c3ffd6b8bb 100644
--- a/client/rest/src/test/java/org/elasticsearch/client/RestClientBuilderIntegTests.java
+++ b/client/rest/src/test/java/org/elasticsearch/client/RestClientBuilderIntegTests.java
@@ -30,6 +30,7 @@
 
 import javax.net.ssl.KeyManagerFactory;
 import javax.net.ssl.SSLContext;
+import javax.net.ssl.SSLHandshakeException;
 import javax.net.ssl.TrustManagerFactory;
 import java.io.IOException;
 import java.io.InputStream;
@@ -37,7 +38,7 @@
 import java.net.InetSocketAddress;
 import java.security.KeyStore;
 
-import static org.hamcrest.Matchers.containsString;
+import static org.hamcrest.Matchers.instanceOf;
 import static org.junit.Assert.assertEquals;
 import static org.junit.Assert.assertThat;
 import static org.junit.Assert.fail;
@@ -72,9 +73,6 @@ public static void stopHttpServers() throws IOException {
     }
 
     public void testBuilderUsesDefaultSSLContext() throws Exception {
-        assumeFalse("Due to bug inside jdk, this test can't momentarily run with java 11. " +
-                "See: https://github.com/elastic/elasticsearch/issues/31940",
-            System.getProperty("java.version").contains("11"));
         final SSLContext defaultSSLContext = SSLContext.getDefault();
         try {
             try (RestClient client = buildRestClient()) {
@@ -82,7 +80,7 @@ public void testBuilderUsesDefaultSSLContext() throws Exception {
                     client.performRequest(new Request("GET", "/"));
                     fail("connection should have been rejected due to SSL handshake");
                 } catch (Exception e) {
-                    assertThat(e.getMessage(), containsString("General SSLEngine problem"));
+                    assertThat(e, instanceOf(SSLHandshakeException.class));
                 }
             }
 
diff --git a/x-pack/qa/smoke-test-plugins-ssl/src/test/java/org/elasticsearch/smoketest/SmokeTestMonitoringWithSecurityIT.java b/x-pack/qa/smoke-test-plugins-ssl/src/test/java/org/elasticsearch/smoketest/SmokeTestMonitoringWithSecurityIT.java
index 52c2a7dfa2d2d..c427d8bf32c86 100644
--- a/x-pack/qa/smoke-test-plugins-ssl/src/test/java/org/elasticsearch/smoketest/SmokeTestMonitoringWithSecurityIT.java
+++ b/x-pack/qa/smoke-test-plugins-ssl/src/test/java/org/elasticsearch/smoketest/SmokeTestMonitoringWithSecurityIT.java
@@ -127,7 +127,6 @@ private boolean getMonitoringUsageExportersDefined() throws Exception {
         return monitoringUsage.get().getExporters().isEmpty() == false;
     }
 
-    @AwaitsFix(bugUrl = "https://github.com/elastic/elasticsearch/issues/31940")
     public void testHTTPExporterWithSSL() throws Exception {
         // Ensures that the exporter is actually on
         assertBusy(() -> assertThat("[_http] exporter is not defined", getMonitoringUsageExportersDefined(), is(true)));