Ask HN: Should I hijack Dropbox and Google Drive for my own app?

I'm creating a web app that includes online storage for users (as well as other features). The storage is obviously going to cost me and therefore I'll have to make it a paid for service in some way.

However, I'm considering simply asking the user to authorise my app to use either Dropbox or Google Drive when they signup and remove the need for me to host their files, yet still have many Gigs of storage available to my app.

Does anyone see any drawback with this. I do recognise that it introduces an extra step for the user. However, if it means I don't have the extra expense of hosting many Terra-bytes of files, then I can make it completely free.