Я выполнил задачу, но не понял почему не было выброшено исключение? 
throw new RuntimeException("UnsupportedOperation!");
Код Solution: 
try (FakeConnection fakeConnection = dbConnectionManager.getFakeConnection()) {
            System.out.println("Entering body of try block.");
            fakeConnection.usefulOperation();
            fakeConnection.unsupportedOperation();
        } catch (Exception e) {
}
Код FakeConnection: 
public class FakeConnection implements AutoCloseable {

    public FakeConnection() {
        System.out.println("Establishing database connection...");
    }

    public void unsupportedOperation() {
        System.out.println("Operation is not supported yet!");
        throw new RuntimeException("UnsupportedOperation!");
    }

    public void usefulOperation() {
        System.out.println("Executing useful operation.");
    }

    @Override
    public void close() throws Exception {
        System.out.println("Closing database connection...");
    }
}
Почему в консоли вывод вот такой: 
Establishing database connection...
Entering body of try block.
Executing useful operation.
Operation is not supported yet!
Closing database connection...
А не вот такой: 
Establishing database connection...
Entering body of try block.
Executing useful operation.
Operation is not supported yet!
UnsupportedOperation!
Closing database connection...