CiviCRM integration with bespoke custom CMS?
It's possible that you could build an integration component to make it work with your custom cms, but it probably wouldn't be worth the effort.

Your best bet would probably be to use the api to accomplish an integration on the pieces you care about.  The api includes methods for most functions, including inserting, updating, deleting and retrieving data using php, rest, smarty and javascript interfaces.  As well as html/js snippets for forms to be embedded into your cms html.  See:
http://dmaster.demo.civicrm.org/civicrm/api#explorer
log in as demo/demo.