Yes! You can download files & attachments related to the emails within a case.
Please follow the below steps to download files & Attachments from "Email Message"
1. Open the "Mass File Download" Tab and Click on "Related Files & Attachments"
2. Now select the "Email Message" from Select Object drop-down
3. Select Advanced Search (SOQL) search mode and hit the Search Button, As shown below.
4. Now hit the Download Files & Attachments button and download up to 2000 files at one time and then another 2000 files and so on