diff --git a/hibernate-core/src/test/java/org/hibernate/orm/test/annotations/entity/BasicHibernateAnnotationsTest.java b/hibernate-core/src/test/java/org/hibernate/orm/test/annotations/entity/BasicHibernateAnnotationsTest.java
index f189d36938..58f75e5341 100644
--- a/hibernate-core/src/test/java/org/hibernate/orm/test/annotations/entity/BasicHibernateAnnotationsTest.java
+++ b/hibernate-core/src/test/java/org/hibernate/orm/test/annotations/entity/BasicHibernateAnnotationsTest.java
@@ -14,10 +14,12 @@ import java.util.List;
 import java.util.Locale;
 import java.util.Set;
 
+import jakarta.persistence.RollbackException;
 import org.hibernate.Hibernate;
 import org.hibernate.Session;
 import org.hibernate.Transaction;
 import org.hibernate.community.dialect.DerbyDialect;
+import org.hibernate.dialect.CockroachDialect;
 import org.hibernate.dialect.OracleDialect;
 import org.hibernate.dialect.PostgreSQLDialect;
 import org.hibernate.dialect.SybaseDialect;
@@ -117,8 +119,14 @@ public class BasicHibernateAnnotationsTest extends BaseCoreFunctionalTestCase {
 			parallelTx.commit();
 			fail( "All optimistic locking should have make it fail" );
 		}
-		catch (OptimisticLockException e) {
-			if ( parallelTx != null ) parallelTx.rollback();
+		catch (Exception e) {
+			if (getDialect() instanceof CockroachDialect) {
+				// CockroachDB always runs in SERIALIZABLE isolation, and throws a RollbackException
+				assertTrue( e instanceof RollbackException );
+			} else {
+				assertTrue( e instanceof OptimisticLockException );
+			}
+			parallelTx.rollback();
 		}
 		finally {
 			parallelSession.close();
diff --git a/hibernate-core/src/test/java/org/hibernate/orm/test/batch/BatchOptimisticLockingTest.java b/hibernate-core/src/test/java/org/hibernate/orm/test/batch/BatchOptimisticLockingTest.java
index b8014ce734..f299c62442 100644
--- a/hibernate-core/src/test/java/org/hibernate/orm/test/batch/BatchOptimisticLockingTest.java
+++ b/hibernate-core/src/test/java/org/hibernate/orm/test/batch/BatchOptimisticLockingTest.java
@@ -10,6 +10,7 @@ import java.util.concurrent.ExecutionException;
 import java.util.concurrent.ExecutorService;
 import java.util.concurrent.Executors;
 
+import jakarta.persistence.RollbackException;
 import org.hibernate.cfg.AvailableSettings;
 import org.hibernate.dialect.CockroachDialect;
 
@@ -29,7 +30,7 @@ import static org.junit.Assert.fail;
 /**
  * @author Vlad Mihalcea
  */
-public class BatchOptimisticLockingTest extends
+public class  BatchOptimisticLockingTest extends
 		BaseNonConfigCoreFunctionalTestCase {
 
 	private final ExecutorService executorService = Executors.newSingleThreadExecutor();
@@ -94,17 +95,18 @@ public class BatchOptimisticLockingTest extends
 			} );
 		}
 		catch (Exception expected) {
-			assertEquals( OptimisticLockException.class, expected.getClass() );
 			if ( getDialect() instanceof CockroachDialect ) {
 				// CockroachDB always runs in SERIALIZABLE isolation, and uses SQL state 40001 to indicate
-				// serialization failure.
-				var msg = "org.hibernate.exception.LockAcquisitionException: could not execute batch";
+				// serialization failure. The failure is mapped to a RollbackException.
+				assertEquals( RollbackException.class, expected.getClass() );
+				var msg = "could not execute batch";
 				assertEquals(
-						"org.hibernate.exception.LockAcquisitionException: could not execute batch",
+						msg,
 						expected.getMessage().substring( 0, msg.length() )
 				);
 			}
 			else {
+				assertEquals( OptimisticLockException.class, expected.getClass() );
 				assertTrue(
 						expected.getMessage()
 								.startsWith("Batch update returned unexpected row count from update 1 (expected row count 1 but was 0) [update Person set name=?,version=? where id=? and version=?]")
diff --git a/hibernate-core/src/test/java/org/hibernate/orm/test/mapping/naturalid/compound/CompoundNaturalIdTest.java b/hibernate-core/src/test/java/org/hibernate/orm/test/mapping/naturalid/compound/CompoundNaturalIdTest.java
index 5796cc1e20..f034acba09 100644
--- a/hibernate-core/src/test/java/org/hibernate/orm/test/mapping/naturalid/compound/CompoundNaturalIdTest.java
+++ b/hibernate-core/src/test/java/org/hibernate/orm/test/mapping/naturalid/compound/CompoundNaturalIdTest.java
@@ -6,6 +6,7 @@ package org.hibernate.orm.test.mapping.naturalid.compound;
 
 import org.hibernate.annotations.NaturalId;
 import org.hibernate.cfg.AvailableSettings;
+import org.hibernate.dialect.CockroachDialect;
 import org.hibernate.query.criteria.HibernateCriteriaBuilder;
 import org.hibernate.query.criteria.JpaCriteriaQuery;
 
@@ -15,6 +16,7 @@ import org.hibernate.testing.orm.junit.ServiceRegistry;
 import org.hibernate.testing.orm.junit.SessionFactory;
 import org.hibernate.testing.orm.junit.SessionFactoryScope;
 import org.hibernate.testing.orm.junit.Setting;
+import org.hibernate.testing.orm.junit.SkipForDialect;
 import org.junit.jupiter.api.BeforeAll;
 import org.junit.jupiter.api.Test;
 
@@ -41,6 +43,10 @@ import static org.junit.jupiter.api.Assertions.assertTrue;
 		}
 )
 @SessionFactory
+@SkipForDialect(
+		dialectClass = CockroachDialect.class,
+		reason = "On CockroachDB the difference between simple and compound natural id is very high"
+)
 public class CompoundNaturalIdTest {
 
 	private static final int OBJECT_NUMBER = 2000;