While I love Tim Draper and the Bitcoin ties he wears, I don't think that will realistically happen in five years. I do think this is possible though if Bitcoin pops all the other financial bubbles (Stocks, bonds, real estate, credit, etc) and causes a massive influx of capital into the network, causing Fiat to become worthless in the process. What would this event be called? Global hyperbitcoinization? If that happens, Bitcoin would definitely become bigger than the internet. Wait... Is it even possible for Bitcoin to become bigger than the internet when it is on the internet itself? This is getting too inceptiony...
nope
physics
bitcoin needs the internet. if bitcoin was to become bigger than the internet then it would end up utilising every website, every IP and a new international communications platform would be needed to cope with bitcoins communication overflow.
also not everyone with the internet will have bitcoins.
bitcoin can become bigger than many services that utilise the internet.
analogy
you cannot have a bedroom thats bigger then the house its built within.. however it can be the biggest bedroom in the house
I don't think OP is asking if Bitcoin can physically and technically be bigger than the internet, he's asking if Bitcoin will have a larger, more profound effect on society like the internet did.