  We study the enumerativity of Gromov-Witten invariants where the domain curve
is fixed in moduli and required to pass through the maximum possible number of
points. We say a Fano manifold satisfies asymptotic enumerativity if such
invariants are enumerative whenever the degree of the curve is sufficiently
large. Lian and Pandharipande speculate that every Fano manifold satisfies
asymptotic enumerativity. We give the first counterexamples, as well as some
new examples where asymptotic enumerativity holds. The negative examples
include special hypersurfaces of low Fano index and certain projective bundles,
and the new positive examples include many Fano threefolds and all smooth
hypersurfaces of degree $d \leq (n+3)/3$ in $\mathbb{P}^n$.
