Think about what a Chromebook would be capable of with always-on 700Mbps.
They could sell that thing with practically nothing inside -- no hard drive, barely any RAM, no processor to speak of, just a screen and a keyboard -- and it could outperform the best desktops that money can buy, just by offloading computation to Google's massive data centers in real time.
Of course, in the short term latency will still be an issue. So more realistic would be to just get rid of the hard drive (why bother? It's practically the same delay to access the nearest data center's SSD as your own), but keep the RAM and GPU, and enough of the CPU to keep things running smoothly. But I definitely expect to see computation offloaded to the cloud as Internet speeds ramp up. It would be super cool to be writing a python script to test something out and just type in a call MapReduce or whatever on gigs of data, and have it Just Work in the cloud in real time.