Not sure if this is an example or not - and I can't remember the source anyway. There was the case of a person calling a 911 operator, and said he was being attacked by an animal in the mountains. Since he lived in an area with no mountains (and because of his slurred speech), the 911 operator realized he was high on something, but she had no way of tracing his location. He ended up getting hit by a car and killed, so this is an example of a cell phone NOT failing but still being useless.
I'm not sure where this would fall but I think this should be included. This is a "super-cut" of all the instances in which the cell phone dies at an inopportune time: https://www.youtube.com/watch?v=XIZVcRccCx0
Not sure where to add this but we should probably mention somewhere on here that most nations have laws to try and avert this. All carriers are required to accept emergency calls from any phone regardless of if it's in their network. Meaning you can frequently make a 911, or local equivalent, call with no bars.