Let's say you have an office on a Gigabit fiber connection going to a network switch capable of processing 10gbe (assume 10x speed). Hypothetically, would the switch be able to respond quickly enough such that even if the gigabit WAN connection was saturated it would degrade performance? It's it just a matter of computing power to deal with the network traffic or do you actually need to have bigger network pipes than the attack bandwidth?
Potentially, your equipment could be incapable of handling packets at line rate (which is more difficult if the packets are small). That's fairly easy to solve though, especially if you're only looking at gigE -- get better hardware and/or software
The problem is if your attacker is sending you more traffic than your incoming bandwidth. Packets will be dropped, and in most cases you won't be able to control which ones. Depending on how the other side is configured, what packets you do get could be highly delayed. That means actual connections to you are likely going to see a lot of retransmits to you as well as from you. It's possible to still make some progress in these conditions, but not very much, processing power won't help.