I encountered something similar a few days ago when doing changes on my custom theme.
If you put a file directly in the folder of your theme
themes
|_ bt
|_ contact-me.php
The contact-me.php structure
<?php defined('ABSPATH') || exit;
// Template name: Contact me
It will be available
If you put it into a folder directly in your theme
themes
|_ bt
|_ pages
|_ contact-me.php
It will be available
Apparently any deeper than that and wordpress can’t find it, so if you have a folder structure similar to this.
themes
|_ bt
|_ pages
|_ contact
|_ contact-me.php
It wont be available.