I had to vote no, i've never cut or nicked my climbing line, thankfully. I've nicked two lanyards and a couple rigging ropes though.
I actually learned how important saw control in a tree is very early in my career. Was a first year climber, cocky as hell. Was on the ground running a rope for a veteran climber and was watching him when he cut himself out of the tree. Fifteen years later and i still can't believe how quick it happened, and it's still high on the list of scariest things i've seen (or heard for that matter.) He landed really bad on a wood pile like six feet from where i was standing. I still vividly remember the horror as i just stared at him for a second, draped over the woodpile, looked like his back was broken, blood running out of his mouth. Almost went and found another line of work that afternoon, and a lot of the cocky disappeared. That image of him still pops in my head when i'm going to be cutting close to my climbing lines and i take the half second to check they're clear before squeezing the throttle.
The climber ended up making it, although he never climbed again, and spent the next couple years working the ground before he retired.