Yes, to answer your question as it is: It is possible.
If you’re also looking to find out HOW, you would need to tell us more information about your hosting though. Do you have ssh access, FTP acess? Is it shared hosting or do you have full access?
If you have SSH here’s what I do:
IF you can use SCP:
Syntax:
scp <source> <destination>
To copy a file from B to A while logged into B:
scp /path/to/file username@a:/path/to/destination
To copy a file from B to A while logged into A:
scp username@b:/path/to/file /path/to/destination
If your OLD server is still live but scp isn’t working:
Login to that server via ssh, make a backup of the directory you want to copy. Ie if you want to copy your www/wp-content/uploads/ directory, use these commands:
cd www/wp-content/
zip -r uploadsbak.zip uploads
now move that file to your webroot folder (www/) either by dragging in ftp or you can just mv it.
mv uploadsbak.zip ../
now login to your new server and request the file:
cd /www/wp-contents
wget www.oldserveraddress.com/uploadsbak.zip
then the wait for the new file to transfer to the new server.
After it’s done, unzip it. and then remember to delete the zip file.