diff --git a/hibernate-core/src/test/java/org/hibernate/orm/test/customsql/CustomSqlOverrideTest.java b/hibernate-core/src/test/java/org/hibernate/orm/test/customsql/CustomSqlOverrideTest.java
index f6c6e2b652..4547e31b18 100644
--- a/hibernate-core/src/test/java/org/hibernate/orm/test/customsql/CustomSqlOverrideTest.java
+++ b/hibernate-core/src/test/java/org/hibernate/orm/test/customsql/CustomSqlOverrideTest.java
@@ -27,7 +27,7 @@ import static org.junit.jupiter.api.Assertions.assertNotNull;
 @DomainModel(annotatedClasses = CustomSqlOverrideTest.Custom.class)
 @RequiresDialect(H2Dialect.class)
 @RequiresDialect(MySQLDialect.class)
-@RequiresDialect(PostgreSQLDialect.class)
+@RequiresDialect(value = PostgreSQLDialect.class, majorVersion = 13)
 @RequiresDialect(SQLServerDialect.class)
 public class CustomSqlOverrideTest {
     @Test
diff --git a/hibernate-core/src/test/java/org/hibernate/orm/test/locking/OptimisticAndPessimisticLockTest.java b/hibernate-core/src/test/java/org/hibernate/orm/test/locking/OptimisticAndPessimisticLockTest.java
index 31d9b9fb14..3c45438a3a 100644
--- a/hibernate-core/src/test/java/org/hibernate/orm/test/locking/OptimisticAndPessimisticLockTest.java
+++ b/hibernate-core/src/test/java/org/hibernate/orm/test/locking/OptimisticAndPessimisticLockTest.java
@@ -5,7 +5,9 @@ import static org.assertj.core.api.Assertions.assertThat;
 import java.util.stream.Stream;
 
 import org.hibernate.LockMode;
+import org.hibernate.dialect.CockroachDialect;
 
+import org.hibernate.testing.SkipForDialect;
 import org.hibernate.testing.TestForIssue;
 import org.hibernate.testing.orm.junit.DomainModel;
 import org.hibernate.testing.orm.junit.SessionFactory;
@@ -23,6 +25,7 @@ import jakarta.persistence.Version;
 })
 @SessionFactory
 @TestForIssue(jiraKey = "HHH-16461")
+@SkipForDialect(value = CockroachDialect.class, comment = "CockroachDB uses SERIALIZABLE isolation, and does not support this")
 public class OptimisticAndPessimisticLockTest {
 
 	public Stream<LockMode> pessimisticLockModes() {