This is exercise 4.19 in Davey and Priestley, Introduction to lattices and order, 2nd ed.
Show that, for a distributive lattice $L$ the following are equivalent:
(1) $L$ is finite;
(2) $L$ has finite length;
(3) The set $J(L)$ of join-irreducible elements of $L$ is finite.
I can prove (and I did it) that (1) and (2) are equivalent and each implies (3), but (3) doesn't seem to imply the others. For example, the lattice $\mathbb{Z}\times\mathbb{Z}$ with the order of the direct product, where each factor has the usual order, is infinite and distributive, but it doesn't have join-irreducibles, and so $J(\mathbb{Z}\times\mathbb{Z})=\emptyset$, which is finite.
Is my thinking about this right?