I am working on a Wordpress-based project, and I just added a new template to the installation by uploading a template file. I was creating a new page that will use my new templ
There are a lot of great answers on this question, but one obvious solution, which initially escaped me, was to double-check that in Dashboard > Settings > Reading you want to set "Your homepage displays" to "Static Page" and then leave the "Posts Page" unselected.
While themes and a properly coded theme file play a role in the situation, the reason that the page attributes area on the edit page screen doesn't show the templates drop-down may be because you selected a particular page to be your "Posts Page", and so WordPress is not asking you for a custom template to use because it's already using front-page.php or home.php (See WordPress Template Visual Hierarchy).
Hope this helps someone like me who had a properly coded template file but didn't realize this setting had been selected.