Can't add submenu

I am using an older theme and have added the following code in my header.php template file to enable custom menus:

<!--This code added for WordPress 3 menu support --><?php wp_nav_menu( array( 'theme_location' => 'header-menu' ) ); ?>

In my functions.php file I also added the following:

<?php
function register_my_menu() {
  register_nav_menu('header-menu',__( 'Header Menu' ));
}
add_action( 'init', 'register_my_menu' );
?>

The menu works great except for one thing: I cannot add submenu items. Is there anything I can add to these snippets to make the menu add child items?