On CAT 5e, you can do 1000 Mbps over 100 meters.
1,000 Mbps = 60,000 MB per minute = 125 MB / second
On my Dlink DGL-4300 router I get around 890-924 Mbps on speed tests. It is still beyond the 24 Mbps of a tape drive. My Exabyte VXA-320 maintained 720 MB/min (12Mbps) across the router with no problem.
Unless you are backing up over miles you don't need fiber for a backup. CAT 5e copper is sufficient for backup jobs.
If you bandwidth is filled with other traffic, that could be an issue. You might need a larger pipe or second backup network. I used to put a second subnet just for backups with 1 Gbps NICs on CAT 5e to a 12 port 1 Gbps full duplex switch. No routing between NICs on each box. Users on their subnet and backups on 1 Gbps network.