The point at which you need to worry is when a company believes it doesn't need a DBA. It doesn't have to be the reality, just the belief.
If a DBA has set up their servers correctly then they will have automated a whole lot of their daily tasks. You can reach the point where an instance can be left to its own devices for a considerable period of time before some action is needed. I'd say that the period of time will be until the people who got rid of the DBAs have been congratulated, been promoted, moved on.
Things won't go pear shaped overnight. The decay will most likely be the slow erosion of security. More users granted elevated privileges etc.
Service packs not applied. I doubt that the lessons of this weekends NHS debacle will be learned.
Eventually someone will do something in a production environment that requires a full restore. At that point they will find that the backup jobs have been failing for several months and there is no backup.