Yes! The internet DOES slow down as the number of ACTIVE users increases. Let me explain. There are SERVERS and NETWORK PATHS involved to realize "internet". While it is true that the world can simply keep adding capacity in servers or networks paths, we tend to see slow-downs when a network path is undersized. This happens when a system built before accounting for future needs such as streaming video, peer-to-peer networks (LIMEWIRE), Xbox Live Gaming, Netflix download, Amazon video on demand, internet radio, etc.
The providers of content can ADD servers to their side of the internet, but unless the network path providers ALSO add capacity, the "internet" will be perceived for any given user as slowing down. The network paths are harder because the path from YOUR house to any given server is probably made up on over a DOZEN sub-segments, owned by many different suppliers. ALL have to be increased to allow the total network path to have the capacity YOU need.
Just because CBS News or CNN adds a bunch of new servers, does mean you'll be able to connect to them from YOUR house as fast as you used to. You are also contending for the bandwidth on the network paths in your neighborhood, your city, your state, etc. It's kind of like a tree structure, with the twigs feeding into branches, and the branches into trunks, etc.
It is VERY common for the internet TODAY to be "slow". Try watching the Victoria Secret fashion show via the web, and you'll see the situation where the SERVERS are overloaded. Now try to surf the web Christmas Morning about 10am and you'll see the web running slow because of all the NETWORK PATHS being overloaded. BOTH will yield a similiar result. All of the major carriers (Comcast, ATT, Rogers, TWC, etc) are modifying their Terms of Usage and Pricing schedules to allow them to develop technology to meter each person's usage, and potentially charge for higher usage in the near future. The goal is to LIMIT the unbridled usage of these network paths.