You could also have the client run some tests and auto report problems such as the bad server's id.
You could also have a manual report button the user can click on. And if you want to get really advanced, have a system that learns from those manual reports so that it can later warn when there is a high probability of a "problem" occurring.
One of my colleagues designed and implemented a telemetry data tagger that allows service owners to annotate the data reported into our primary telemetry system Atlas [0] and has further worked on mechanisms for service owner feedback to automatically create training data sets for this implementation.
Great question, and I'm glad others are thinking along these lines -- we fully believe that this is important for bridging the gap between service owners and our analytics.
You could also have a manual report button the user can click on. And if you want to get really advanced, have a system that learns from those manual reports so that it can later warn when there is a high probability of a "problem" occurring.