Depends on what you have for a processor. Once you have more power than it takes to handle the max bandwidth, it makes little difference how much extra you have, it's not going to be the limiting factor anymore. Today's CPUs can easily handle the max speed of your internet connection - your limitaion is your com link.
I would mostly agree, however there are a couple of other issues that could affect your apparent download speed. These would each be generally extreme issues, should not occur on a machine given even the most cursory attention, and are generally easy to resolve, but they can happen.
If your hard drive is both fragmented and loaded to the hilt, files you download may not be written to the disk as quickly as they could be downloaded. If you've got a ton of background operations running (as in double-digits), it is possible for there not to be enough processor power to move the data around. If those background operations are using the Internet connection, well, you may be getting the speed, but it may be going to operations other than what you expect. Similarly, if you're using a USB connection, if you're printing to your USB printer, and pulling pictures off your camera or scanner at the same time, your Internet download speed will suffer.
In other words, after a round of (what should be routine) maintenance, and some common sense in what you expect in the line of multitasking, even an old Pentium 133 is more than fast enough to not affect download speed. It's certainly possible to create a situation where even the newest, fastest CPU becomes the bottleneck, but I'd suggest that it's not the processor speed that's the problem. The user would be the problem causing the bottleneck.
The quick answer is that it can but there are too many other variables to say.
If you have any reasonably recent computer and don't have additional software overhead (software firewall, virus scanning, VPN, etc.) then you are unlikely to hit limits imposed by the CPU. If you do have significant software the is involved, then you may easily be at a point where a faster CPU would help.