I had the exact same problem: worked flawlessly on local and remote WiFi as well as over 3G after setting up port forwarding on my router, but after a day picture became choppy despite audio streaming uninterrupted. I was able to resolve the video streaming issues by simply resetting the Slingbox. I unplugged the power to the Slingbox (HD Pro in my instance), waited about 20 seconds, and then plugged it back in. After the Slingbox finished booting back up and both lights on the front of the box were a solid red I tried viewing and it worked fine.
Hope this helps and go Red Sox!