I'm not sure this counts as dumb luck, but I went to try and catch the legendary birds on Fire Red. I tried Zapdos first, and had only 23 ultra balls. I ran out 3 times, before I finally caught it on like the 20th thrown ultra ball the 4th time. (It was red and paralyzed all 4 times.) I then proceeded to stock up on like 70 ultra balls for Moltres, so I wouldn't have to start over 3 times, but I caught it in my first ultra ball. >__>
I also found a shiny Electrike on Sapphire, and only had 1 great ball, and I caught it. I found a Shiny Bellsprout in the Safari Zone, and caught it. (with a safari ball obviously. I was so afraid it'd flee. <_<)