Downloading doesn't require the use of Adobe Reader so I'm guessing there may be some more details that you could supply?
How are you downloading?
Are you getting an error during the download (in other words, what exactly happens)?
Are you using a browser? If so, which one?
In case it does have something to do with Reader, it would be helpful to know the exact version you have.
Might also be helpful if you could post a link to the site with the problem PDF's.