How to open PDF from Notification bar and fix "Cannot open file" error?
It's a combination of a PDF and a browser issue.  Some PDFs work fine while others do not across different browsers.

Downloading the PDF you mentioned, here are the results using different browsers:


Chrome: Cannot open PDF from notification
Boat Browser: Cannot open PDF from notification
Dolphin: Can open PDF from notification


Downloading a different PDF from another site, here are the results using different browsers:


Chrome: Can open PDF from notification
Boat Browser: Can open PDF from notification
Dolphin: Can open PDF from notification


UPDATE 2013-10-11: According to Scott's findings, this is a PDF issue. The Content-Type header of the problematic PDF is application/x-download, while the working PDF is application/pdf.

If you control the server which is serving these PDFs, you can switch the Content-Type header to allow it to open in all browsers.  PHP example:

In PHP this was previously broken when sending:

 header('Content-Type: application/x-download');


Switch this to this header to work:

 header('Content-Type: application/pdf');