No, you cannot place these files on a network drive.
The database files (brd) need to remain on your local computer's hard drive.

These flat file databases are fast, light-weight, easy to use and backup - but their locking system is not designed to placed on a storage device managed by a system other than the local operating system. 

Placing these files on a local area network (server) drive or a remote (ie. cloud storage drive) will result in file corruption.

Backups and 'pushing' a copy to a co-worker
If your goal was to have a backup of these files - or to 'push' a copy of the data file to a co-worker:
--Please use our backup feature (File > Backup) and point to a remote/server drive. 
This will 'push' a copy of your file to the specified location.

DropBox, Box, Egnyte and One-Drive

Note: Do not use DropBox, Box, Egnyte, One-Drive or similar service as this will also cause file corruption.
Rather hear it from them?  
https://www.dropboxforum.com/t5/Dropbox-files-folders/Sharing-Access-database/td-p/148125

Did the above not address your issue - and you need true multi-user access?
You may need to upgrade to one of our multi-user platform, such as our Workgroup platform (or possibly Enterprise).

Subscription license upgrade to Workgroup ($40.00/month additional):
https://store.treasurysoftware.com/ProductDetails.asp?ProductCode=UPGSUB40PlatformUpgrade

Permanent license upgrade to Workgroup ($997.50 one-time additional, plus increase in annual support of approx $185 ):
http://store.treasurysoftware.com/product_p/upgprd1000.htm