Umm, I may have misunderstood a couple things, but it sounds to me like steam is trying to update something. It could be a simple steam client update, in which case the only thing you can do is let it finish. (I don't think any steam client updates have been release lately unless you haven't been on in a while in which case you could be getting an old update).
The other issue might be that steam is automatically updating 1 or more of your games, or perhaps your Source SDK if you have it.
Every time I get problems like the one you're describing, it turns out Steam is the cause.
Also, when this happens, jump out of game and steam and see if you can use the internet for browsing web pages. If everything seems to be going at the normal speed once you exit steam, then it's probably safe to say steam is the problem.