Disclaimer first: The brake-hand principle (always hold the braking strand of the rope) always applies to assisted braking belay devices, including the Grigri. This question is in no way targeted at putting that in question, follow it at all times!
In a recent question about quickly belaying on top rope, some discussion happened in comments about how the braking mechanism of a Grigri can fail. In general, the Grigri is known as a very reliable, while bulky tool for belaying with a brake-assistance.
Now excluding user failure (e.g. fully holding the Grigri in one's hand and thus disabling the braking mechanism), what scenarios are there in which the Grigri will not by itself start braking and finally arresting a fall?
In addition: What is/are the countermeasure(s) to let the braking mechanism engage in the respective scenarios?
I am only looking for scenarios which can be/have been demonstrated (in practice or in a test setup), not in hypothetical problems.