Assume that a server has Full Disk Encryption enabled - LUKS for Linux and BitLocker for Windows. This FDE requires a passphrase on boot to allow the disk to be unlocked and facilitate the boot process.
Many pieces of Backup and Recovery software offer the option to encrypt your backup. Presumably this prevents a attackers ability to mount your backups a la Phineas Fisher should they fall into the wrong hand or you have an actor in your network by requiring a passphrase to decrypt the password.
Assuming your encryption passphrases for your FDE is different from other passwords on your network to eliminate password replay attacks, does encrypting your backup in addition to using FDE offer any additional protection or is it simply overkill?
EDIT: My question is in reference to strictly technical security, and procedural issues / people issues should not be taken into consideration.