Are there any CMS's out there that allow you to upload your own files you created in Sublime?

My own HTML and CSS3 files etc

I am not sure I full understood what you are looking for but for starters, I will assume in the sense of a theme as most CMS's will accept custom themes as long as they are designed as per their documentation for example if you want to make one for wordpress you will have to use PHP together with html, css etc to make a theme for it. You can check some documentation of some of the CMS's here you can research for more of course:

WordPress Joomla Drupal

Now if you mean upload the files of projects codes you could use a tool as github, bitbucket etc. You can have a private repository (not free) or a public one.

You could also upload your code on your server as a back up or on a cloud service (if you use a cloud service you can give access to the code to your client through a link).

These are some suggestions.

I think something like Couch is exactly what you're looking for (super simple cms): http://www.couchcms.com/

"Take any HTML/CSS template and make it CMS enabled in minutes. Just drop in a few special XHTML tags into your code and watch your static web pages come to life!"