It seems to me that for a file server the main bottleneck/performance concern is going to be hard disk speeds rather than cpu. Unless your IO devices use a significant amount of CPU (think older drives in PIO mode instead of UDMA). I'm not sure what sort of CPU usage a RAID array causes, nor am I sure what CPU usage a network attached storage device uses.
If we imagine a simple FTP server, for example. Typical architecture for that would be to have a main thread of execution "listening" for connections on a specific address and port. When a connection arrives a sub-thread is created (or re-used) and in this thread the connection is handled.
I imagine copying files using windows explorer has a similar architecture to what I described for the FTP server. The main process explorer.exe handles the copy requests but creates sub threads to handle the actual copy of data.
Therefore you could say that generally a multi-core CPU will have an advantage over a single core system simply because it can handle more connections simultaneously, devoting more CPU time to each.
That said, if the speed of the hard disks, or speed of the network connection are slow enough then these bottlenecks will prevent the cpu from ever reaching 100% usage, and in such a case the extra cores wont benefit you much.
So, if your IO devices require a lot of CPU to operate at full capacity, or your IO and network speeds and high, then you will want more CPU power in order to reach maximum potential.
If your IO devices require little or no CPU, or your IO or network speeds are low then you will not need as much CPU to reach maximum potential. This will be a lower potential than the high speed, high CPU situation.