When will I need to have any of my skills above 100% to pass skill checks? Being able to continue raising them past 100 is pretty cool, but I'm confused about when that will be important. I know it will always help my chances, but will there ever be any checks I outright can't pass with 100 in a skill? At about what point would y'all recommend I have my skills at?

You are only mediocre at skills at 100, for all intents and purposes, 200 is essentially the equivalent to 100 in the newer fallouts.

Also note that progressively past 100 it costs more and more skill points.

But yeah, the master challenges for skills will require around 180 in a skill (like to get skynet for example, I think you need like 175 science skill).