Yes, you can search for individual records by searching in the Related File & Attachments Tab under the Mass File Download tab.

We have 3 Search Modes after we select a particular Object from the Select Object drop-down list named - 

a) List View

b) Search

c) Advanced Search (SOQL)

Select the Search option and enter more than 2 characters of the record name you want to search in the search box - The Search box will then show a list of records matching with your search term, then you can select the record and click on the Search button.

If the selected record has any files or attachments then it will show all the files and Attachments in the result box like the below image -