This is possible, but it will require playing with the rewrite rules. Install my Rewrite analyzer, you will need it.
I answered a very similar question, but the main difference was that where you have /projects/ as a “clean” custom post type archive, the other user had it as a page, which is much easier.
If you want to create subpages of /projects/ you will have to create a page with the slug projects. Then you have two options: the easy one is to use a page template for this /projects/ page that just queries for your custom post type, like we used to do before WP 3.1 arrived with native support for custom post type archives. The other option is to filter the verbose page rules and throw out everything related to the /projects/ page, so that only the /projects/ rules for the custom post type archive remain.