I think this is common in algebraic geometry too. For example while most curves have dimension of their Brill--Noether space given exactly by the Brill--Noether number $\rho$ (thanks to the Brill--Noether Theorem of Griffiths and Harris), it is hard in practice to write down any particular curve which does.

(This is also an answer to the linked "finding hay in a haystack" question.)

I think this is common in algebraic geometry too. For example while most curves have dimension of their Brill--Noether space given exactly by the Brill--Noether number $\rho$ (thanks to the Brill--Noether Theorem of Griffiths and Harris), it is hard in practice to write down any particular curve which does.

(This is also an answer to the linked "finding hay in a haystack" question.)

EDIT: And I should say why this shows the non-generic but "natural" curves are "better behaved" is because it means their Brill--Noether spaces are larger than you would expect, i.e., these curves have more "representations" (maps into projective space) than you would guess.