Two examples from lattice theory:
is every lattice with unique complements distributive? [no]
is every distributive algebraic lattice isomorphic to the
lattice of congruences of a lattice? [no]
See http://www.ams.org/notices/200706/tx070600696p.pdf

<cite authors="Grätzer, George">_Grätzer, George_, [**Two problems that shaped a century of lattice theory**](http://www.ams.org/notices/200706/tx070600696p.pdf), Notices Am. Math. Soc. 54, No. 6, 696-707 (2007). [ZBL1286.06001](https://zbmath.org/?q=an:1286.06001), [MR2327971](https://mathscinet.ams.org/mathscinet-getitem?mr=2327971).</cite>