I was the lead contact for a 24 hour emergency support team for a mission-critical software system that my company developed. Our system MUST NOT be down. For any length of time. And there were potentially human lives at stake. Heh, you know where this is going... =)
The client was on the East Coast and when they started their day at 5am Eastern, they noticed that the system seemed to be down. It was 2am Pacific when I got the call from a very panicked client. I was equipped to do remote systems support, so I logged in and very quickly verified that our system was working just fine; the problem was in another system that was interfacing with our software.
At that point, technically, my job was done. There was really nothing more that I could do. But I recognized that the situation was still not resolved for the client. I stayed on the phone with them, I walked through some basic diagnostics and suggested things they could try to troubleshoot.
I stayed in contact with them throughout the night. I helped them coordinate the effort by updating different people on their side on the status of the situation. I gave them feedback on our system's responses to things they tried. I was generally helping to dissipate the feeling of panic and was there for moral support. I did not leave them until they were satisfied that everything was a-okay again.
The problem was finally resolved to their satisfaction and they were very appreciative of my efforts.