Actually, experts have predicted this to happen around 2010. They say that with increased bandwidth usage, we will experience something like the effect of an internet "traffic jam". Computers will freeze up and we will have brownouts, with periods of frozen and slow internet access. Another contributing factor to the internet getting too full is the lack of available IP addresses (a unique number assigned to every computer).
I guess in theory it is feasible for the internet to get full if we use up all of our resources for transmitting information. The internet is actually a network of computer networks that depend upon its electronic infrastructure to transmit information. So if you think about bandwidth as the internet highway and the ip addresses as the locations of such information, then the question remains "How do we prevent the internet from getting full?"