Let $V$ be a finite-dimensional vector space, let $U_1,\dots,U_n$ be subspaces, and let $L$ be the lattice they generate; namely, the smallest collection of subspaces containing the $U_i$ and closed under intersections and sums. Is $L$ finite?

This is well-known to hold if $n\le3$: if $n=3$ then there are at most $28$ elements in $L$, indpendently of $V$'s dimension.

Note that I suspect the answer to be "no" if $V$ is allowed to be infinite-dimensional: there exist infinite modular lattices generated by $4$ elements; here $L$ is a bit more than modular ("arguesian", see Is the free modular lattice linear?) and finiteness of free arguesian lattices doesn't seem to be known. It would be nice to have an example.