I've never had rental gear problems so bad as to end a dive or a trip, but I have had a couple that made a dive less than pleasant:
On my first OW dives, I forgot ONE fin at home (used it to measure a suitcase and forgot to take it out) and had to rent full-foot fins from the local Nassau dive shop that didn't fit me as well and let in some abrasive sand. Not fun.
And on my most recent trip, I had two problems with my rental reg setup: first, the console computer stopped working on the boat. I already had a wrist computer, so I replaced the computer with an extra SPG from my buddy (friend of mine). Then on one of our dives, my 2nd stage started "wet breathing". The trickle of saltwater leaking in with every breath was so irritating, I swapped my second stage for my octo for the rest of that dive and all of the next, and that evening I replaced it temporarily with a rented second stage from the Turks dive shop. Like the fins on my first few dives, it wasn't ideal, but I was fortunate to have access to a full local dive shop with spare rental gear if we needed it.
The only time I've come close to a dive/trip ender was when a rented octo briefly had a freeflow problem, but I got it under control and didn't have any further trouble. If it had happened again, I might have been out of luck, because on that trip, I was in a remote area with no dive shops around.
I've been steadily buying my own gear, however, and have no problem paying bag fees or toting it halfway around the world, because being familiar with dive gear is an added safety benefit that you don't get from rentals.