Give us an example of how things went wrong for you, please?
Once a long time ago we were in Pennsylvania playing an Octoberfest event. We were in a semi-outdoor environment, a big place with no walls but a high ceiling. It had been raining off and on all day and in the course of a couple of songs while we were performing, it got dark and stormy.
Out of the blue a bolt of lightening hit somewhere nearby and knocked out all power and all electronic amplification and electric instruments went out leaving only the accordion, trumpet, sax and drums being audible. We didn't stop, playing through the song for about a minute when the power came back. The people just started applauding, we finished off the evening without further interruptions. The only change I made was move from supporting instrument to main melody and played as loud as I could, when the organist regained power she took up the melody and I switched back to being harmony.
In that case, we just pushed through it and moved on, that is about all that could be done. I suppose it depends a lot on what happens and you deal with it the best you can. Basically unless it is a catastrophic event, "the show must go on".
This is the nice thing about music, at it's worst, if something happens, no one is hurt and no one dies. The music stops, and if it is possible, it begins again, perhaps after a short break to sort things out, restart things or do what is needed to get things going again.
Another time I was called in to provide some background music for a dinner function. I was very tired, having had a super hard week, and I just blanked out in the middle of a song that I was playing. I just restarted the song intending to push through, crapped out in the exact same place. I smiled, laughed, and moved on to another song. Inside I was mortified and wanted to die, but the world kept moving irrespective and nothing bad happened other than the music stopping twice.
Wow, I had totally forgotten about both of these events, thanks for bringing them back for me!