I'm creating this to hopefully help others that are having this issue. I typically issue Firefox, but had a website that only worked with IE. When I recently upgraded to IE9, I and no longer could open PDF's from that site. This feature was working fine on IE8, but on IE9 I kept getting "This file could not be downloaded". It was actually a secure website using https that this occurred on. I searched for a solution and finally found it on the Microsoft Answers website, but thought it would help to post it here as well.
From your IE9 browser, Tools->Internet Options->Advanced. Scroll down to the security section and remove the check from the 'Do not save encrypted pages to disk'. This solved the issue for me and hopefully it will for you.