I haven't gone as far as making a "responsive" datatable with bootstrap, but the two certainly can be used together.
Check out this link: http://datatables.net/blog/Twitter_Bootstrap_2 and keep in mind the article is for Bootstrap 2. For Bootstrap 3, you will need to refer to http://getbootstrap.com/getting-started/#migration as some class names have changed between Bootstrap 2 and 3. It shouldn't be too difficult from there, if you're familiar enough with Bootstrap, to get a responsive datatable.
Edit: Allan also has some Bootstrap 3 integration files already whipped up here.