I think I have tracked my problem down to Types. But I still can't explain exactly what is happening.
On my live site, I have a main menu. That menu has a menu item that links to a page. The page has a template assigned to it. But for some reason, when I click on the menu item for that page, the index page for the site gets displayed. I was expecting the custom template to get displayed.
My site uses 2 plugins, WPML and Types.
For a test, I setup a local copy of WordPress. I used the (twentyeleven) theme. I just added one file called page-custom.php to the theme. No other modifications were made to the theme. I installed WPML and setup up a page with a custom template, page-custom.php. I "attached" that page to the main menu. When I clicked on that menu item, the custom page template got displayed correctly. I then installed Types. The menu item was still working correctly.
Then I imported the Types data from my live site. The menu item stopped working correctly. When I clicked on it, it only showed the content of the index page. The URL seemed correct meaning that it looked like it was the URL for the page I was trying to display.
On my live site, I used the WordPress administrator user interface (UI) to set up everything for Types.
It looks like my Types data is affecting the display of pages. I don't know why that would be since I used the WordPress UI to set up everything.
Does anyone have an idea of what I could try to fix my issue with displaying a page from a menu?
WPML Multilingual CMS: 2.4.3 (and the latest versions of the related plugins)