  We compare the sets of Calabi-Yau threefolds with large Hodge numbers that
are constructed using toric hypersurface methods with those can be constructed
as elliptic fibrations using Weierstrass model techniques motivated by
F-theory. There is a close correspondence between the structure of "tops" in
the toric polytope construction and Tate form tunings of Weierstrass models for
elliptic fibrations. We find that all of the Hodge number pairs ($h^{1,
1},h^{2, 1}$) with $h^{1,1}$ or $h^{2, 1}\geq 240$ that are associated with
threefolds in the Kreuzer-Skarke database can be realized explicitly by generic
or tuned Weierstrass/Tate models for elliptic fibrations over complex base
surfaces. This includes a relatively small number of somewhat exotic
constructions, including elliptic fibrations over non-toric bases, models with
new Tate tunings that can give rise to exotic matter in the 6D F-theory
picture, tunings of gauge groups over non-toric curves, tunings with very large
Hodge number shifts and associated nonabelian gauge groups, and tuned
Mordell-Weil sections associated with U(1) factors in the corresponding 6D
theory.
