Well basically, it seems to be related to something, or some series of things you do in the game. It manifests itself when you attempt to load a saved game. When you select a saved game it advises that the save relies on download content that has been moved. If you try and load the game, either the load screen appears and freezes, or you are advised that the Storage device is no longer available.
The issue seems to happen to different people at different times. A lot of people who had it happen say it happened after they entered Vault 3. My issue happened when I killed a certain NPC.
Save regularly!