If it is a bandwidth issue, there is probably not much you can do against it (other than upgrade your plan if possible).
Simple steps like stopping any downloads, Windows updates or any other streaming apps while watching videos may help.
Streaming off peak hours will probably leave more bandwidth available to you as well...
As to exchanging your entire system, that is entirely dependent on how you use your computer.
To begin, I think using your current 3870K with a nVidia 640GT makes no sense.
The 3870K's greatest strength is in its excellent IGP, the 6550D.
Using this processor with a dedicated GPU not to much more powerful just seems like such a waste.
That said, the 3870K is an excellent all around processor suitable for light gaming and general workloads.
On the other hand, an Intel build, depending on the specific processor, can be quite a bit faster in highly demanding tasks.
Coupled with the much more powerful 7770 it would also be a much more capable gaming machine.
With all the increased performance, you will be paying quite a bit more and using quite a bit more energy.
So, it really comes down to what you plan to do on your computer.
For standard usage and light gaming, your current AMD build less the 640GT is quite good.
If you want a gaming build, exchanging for a more expensive Intel build is a compelling option.