Are you seeing this on both the simulator and a device? The reason I ask is that I have seen this from time to time, but only on the simulator. My guess is that the simulator may be losing track of what requests are outstanding, but if you see it on a device too then it could be something else.
Also, you mentioned that you see this problem with the stock ArcGIS samples/templates. Is that correct? Can you ensure you don't have any extra calls in your application to show the activity indicator?
You may already be aware of this, but in beta versions of the API, developers had to manage the network activity indicator when kicking off tasks. However, at the final 1.0 version, the tasks manage the indicator for you. Be sure to remove the network indicator related parts when migrating older code.