When a diaphragmatic hernia occurs, some of the abdominal contents push up, right?

Usually: However if the hernia is covered by the liver, the liver is big enough to prevent things from going into the chest.

Yes: A diaphragmatic hernia is a hole in the diaphragm through which abdominal contents herniate in to the chest. Most common is a hiatal hernia, which involves the stomach. Other types include congenital holes, and are called Bochdalek and Morgagni hernias. You could also have a tear caused by trauma. Only treatment is surgical repair. Hope this helps!