When you say it didn't begin, do you mean that it stayed as scheduled / pending until monday morning, or that it was running but didn't actually start backing up data until monday morning?
There are several things which could be causing this. First check that the NIC's in all related servers are set to run at the same speed (I assume 100Mbit). Ideally, turn off auto detect, and set them all to 100Mb Full Duplex, so you can be sure they are all running at the same speed.
Check that nothing new has been scheduled on the server to run at that time which could slow the server down while the backup is running.
Check / change the network cables you're using, in case one of them has been damage somehow.
Finally (or least the last I can think of right now), if possible consider changing the NIC's in the servers to confirm if there is a fault in one of them. We had the same situation on one of our backup systems for ages, which was quite often intermittent, and would clear after a reboot. After a lot of fiddling we eventually found that the NIC was faulty, so had to get it and the motherboard (as it was integrated) changed, but that did resolve the issue.