May NIC bonding on linux media server cause backup failure with network error

i would say try bond mode as mode=1 (active-backup) for one media server and try the backups using the media server, if that works.. then discuss this configuration with Architect or Designer\ systems admins\Network admins then implement it .

you need to make sure that when you are using this mode 1, you need tp balance the interface/network usage across the switches to make sure the load it distributed evenly for all media servers with the help of network admins

For sure if backup run OK without bonding, then bonding IS the problem. 

"Testing of one full backup with one of the media server which does not have bonding was successfull without any retry"

Lot of documentation out there how to configure bonding, so I recommend take a look of the switch vendors documentation and team up with the network admin