This thread on the Apple Support Discussions forums describes the issues. User "brownrecluse" opens the thread:
After installing leopard I cannot open any attachment. I get this error on every email I get with an attachment: Mail was unable to save the attachment "image.jpg" to disk. Verify that your downloads folder exists and is writable.
Of course I checked my downloads folder exists and is writeable, I even tried creating a new folder and get the same error.
Obviously something with the Leopard install because I was downloading attachments right before that.
Please help!! I can't download any of my clients email attachments!!
Attempts to fix the issue do not seem to be working and my own attempt to recreate the problem yielded no results. Users have tried repairing permissions, creating a new Downloads folder and adjusting the permissions, and creating a new mailbox in Mail. An official fix for this has not been relayed by users or by Apple. Other suggestions to help solve this problem are to rebuild your Mailboxes or do a complete reinstall of Snow Leopard. When attempting any of these troubleshooting techniques, please make sure you have a stable backup of all your important data.
A workaround does exist, however. Simply open the message in a different mail client and download the attachment that way--try Thunderbird or the web version of your mail server (e.g., Gmail, Yahoo! Mail, etc.).