I ran this command, echo "10.0.16.12:/mnt/data/backup_radius2 /var/backup nfs defaults 0 0" >> /etc/fstab. Followed by mount /var/backup and got the error mount.nfs: access denied by server while mounting 10.0.16.12:/mnt/data/backup_radius1. I then ran umount 10.0.16.12:/mnt/data/backup_radius1, which worked as far as I can tell. When I ran a cat of /proc/self its no longer showing the radius1 directory to be mounted.

But when I once again run echo "10.0.16.12:/mnt/data/backup_radius2 /var/backup nfs defaults 0 0" >> /etc/fstab followed by mount /var/backup, I still get the same error. Any suggestions?

Welcome to LinuxQuestions.

In my opinion it is a very bad habit to add lines to your /etc/fstab using echo. Hopefully you have backups. One slip up of > versus >> and you have overwritten the file. If you ran the commands twice now you have two lines that are the same.

You posted "echo 10.0.16.12:/mnt/data/backup_radius2..." but mounted 10.0.16.12:/mnt/data/backup_radius1. A typo or different share? How is it defined in your server's /etc/exports file?

there's some confusion between ..._radius1 and ..._radius2 here.
please clarify.
also, if you get nfs permission denied errors, you have to show us the setup of the server as well as the client (and clarify which is which).

I was attempting to mount the radius2 backup share, which is on my NAS server for backup purposes. Radius1 is a separate share on the NAS. I was running these commands on the client.

How is the backup_radius2 share defined on the NAS? Post the contents of the /etc/exports file.