Does a template override require a subtheme? Feels like a silly question to ask, but I can't find a definitive answer anywhere.

Background

My project is a multi-site that features several themes that our users can choose from. These are all installed in /sites/all/themes. For one site in particular, I wanted to simply use a different page.tpl.php file. I tried copying the theme's page.tpl.php file into /sites/my_site/themes, but this didn't seem to work. This leads me to the question:

In order to override a template, must I use a sub-theme?
The correct answer is no since you can also override templates in a module (example). It doesn't sound like you want to do that, though.

At minimum you would have to create an *.info file that includes the dependency:

name = My Theme
description = Just a basic theme
core = 7.x
base theme = base_theme_name