In one of my LeafGreen playthroughs, I was underlevelled for the E4, but I challenged them anyway. I somehow managed to beat them all, so I saved as I was about to challenge the Champion. He totally thrashed me each an every time I tried to beat him. I couldn't escape from it, so my only choice was to let him wipe me out so I could go and do some level grinding.
Also, in one of my Gold playthroughs, I caught a shiny Caterpie, then later on started a new game save... I miss that Caterpie.