What kind of file is the attachment? Firefox should be able to display images and PDFs. When you follow a link to a type of content that Firefox doesn't display in a tab, you usually get a download dialog with Open and Save options. So that wouldn't surprise me if it's a Word/Excel or other document Firefox can't display.

There also may be some situations in which the link protocol is altered. Instead of http:// or https:// it might be something else which Firefox does not recognize. If you hover your mouse over the link and look at the status message in the lower left (sometimes right) corner, do you see anything unusual about the link?