On our last trip, we were at least 10 minutes late to every ADR, a couple as late as a half hour past our time and we managed to get in to every one of them. I don't know if it's the pixie dust or what but this year we're determined to be on time!!
ETA: we showed up 40 minutes early for our ADR at LTT once and had to wait until 15 minutes past our ADR time anyway. That did discourage us from showing up too early.