Dropbox has been prevalent since 2007 however, I have been one of the late users of the service. Dropbox provides storage that is synchronized with your machines (phone, laptop, etc.) with exceptionally simplified user experience.
The service allows a very simple sign up mechanism using any of your email accounts and without having to be referred from someone else. On sign up, it offers you only 3 options, basic- free service with 2 GB space and the other 2 monthly paid options. Being an individual user, I would go for free service. However it is very apparent that 2 GB free space is not enough at all in this digital age.
Therefore, all drop box has to do is wait to get as many customers as possible to be signed up and use free service. Once such customers are hooked up for a longer duration and satisfied with the service, they would be ready to pay for paid service after they use up their free space.
With the free service model (trial model), drop box targets to get maximum number of users to be signed up for free (so far 200 mil) with an assumption that there is enough cash flow without dropping the services provided to consumers.
The model has been successful since it was launched during the time when 2 free GB storage was useful enough for at least 2 years. However, this current time frame, with other competitors in market such as Sky Drive (Microsoft) or Google drive that provide more storage space (up to 7 GB), users would prefer to have an additional account in these services rather than using dropbox just because they would be forced to pay for more space that is freely available with other competitors.