Short answer: yes. That's how many browsers work.
Firefox: http://msujaws.wordpress.com/2012/04/23/an-update-to-site-identity-in-desktop-firefox/
Chrome: why doesn't the favicon file show in the address bar of google chrome? and http://productforums.google.com/forum/#!topic/chrome/kUQOKVUwUmw
In the wikipedia page, you can find some more about browser support. In short, Firefox and Chrome don't do it, Explorer, Safari and Opera do.