What upload speeds do you get when you have a computer connected to the same router as your SB and you visit bandwidth-testing websites such as www.bandwidthplace.com? That might explain the slow speeds, but losing connection altogether could relate to something else, such as the MTU setting on your router/cable modem.
I have not experienced this particular type of problem myself, but I've seen others in other threads describe solutions that involved changing the MTU setting.
What I'm getting from your post, is that you're having problems with the Slingbox connectivity.
In this case, what I would recommend, is to make sure you have a hardwired connection between the Slingbox and router as first step.
Besides that, discard any network or streaming restrictions preventing the Slingbox to work properly.
You can also check the upload and download speed since that could affect the streaming as well