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