I'm thinking about the famous problem from classical algebraic geometry of how many lines in $\mathbb{P}^3$ meet four given general lines. According to some lecture notes on intersection theory that I was reading, Schubert had the intuition that it's general enough to consider the case where the four lines are in fact two pairs of intersecting lines - a highly nongeneric setup. (And then of course, in this case the answer is easily seen to be two.)
My question is, can someone give a basic explanation of this intuition, and perhaps some explanation of which other scenarios admit this sort of logic (i.e. looking at non-generic-but-still-kind-of-generic cases)? I have a vague picture in my head of continuously varying one of the four lines and how a line meeting all four should vary continuously, but I'm not entirely convinced by it yet.
Thanks! (Also thanks to Michael Joyce for pointing me to this sort of problem in a previous answer.)