Since you have a leading / in your href, you are referencing a file that will be in the root-folder. In case you have your page in a folder on your computer, not serving it from a local webserver, the leading / will tell the browser to look in the root folder of your filesystem. So the browser expect the file to be at C:/favicon.ico or similar, which is probably not what you've expected.
If you have the favicon.ico in the same folder as the web page, you could just remove the leading slash, and the icon should be visible.
<link rel="shortcut icon" href="favicon.ico" />
Update:
As a debug option, your could try to add a tag that you know works. I borrowed this snippet from the StackOverflow source. Try replacing your link tag with this and see if you get the SO logo as your favicon.
<link rel="shortcut icon"
href="http://cdn.sstatic.net/stackoverflow/img/favicon.ico">
Update 2:
It appears that there is a bug reported on Chromium where the favicon isn't displayed if the file is loaded locally, without being served through a webserver.
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