If well configured, the system should be fault tolerant (i.e if any node drops off the network there are other copies available elsewhere. Many systems already implement this (including Bitcoin itself). To be more robust than Dropbox, all you have to do is have more system uptime than AWS (or whatever service they're using to store all that data).
One of the things that people tend to underestimate is datacenter outages. When an Amazon datacenter goes down it takes out many sites with it. We're working toward a world where data is redundantly kept in many different datacenters/locations. In those almost-total failure cases, latency is better than no availability.
