Yes, but you would need to do it in another way. When you download the files they have the name within the Mendix database and that database also holds the original name of the file. Either use the SFTP module so you can export all the files to some SFTP server.
You also use one of the zip modules in the marketplace. You could then create a mciroflow that retrieves batches of files, put them in a zip and then directly download that file. if you restore the full database locally and then create those zips they are already in a nice format to hand over to the other party or if this process needs to be done on a regular basis you could make a form where the end user could select the files in the cloud.
Hope this helps a bit.
Although you have 5000 files to download, these are likely only one or a couple of filedocument-specializations. You can create a download function for each of them, using the download widget.