Dropbox actually runs on AWS by using Amazon's S3 for storage. This is what allowed them to get up and running quickly and cheaply without needing a lot of venture funding.
I would not be surprised with Dropbox's continued growth, if they in the future were to set up and manage their own data centers to avoid being dependent on Amazon or anyone else's platform. This would be advantageous if Amazon were to decide to compete directly with a similar product.