The breadth of a lattice is the largest integer $n$ such that any join of elements $X=\{x_1,x_2,\ldots,x_{n+1}\}$ is join of a proper subset of $X$.
Birkhoff's classical book has an exercise: "Show that the smallest lattice with breadth $k$ is the Boolean lattice of $2^k$ elements."
I think that a little more can be said: A lattice with breadth $k$ contains a sublattice isomorphic to Boolean lattice of $2^k$ elements, but not of $2^{k+1}$ elements. But is this false, true and already said somewhere, or true and not already explicitly said?
The answer is yes and already said. Varieties generated by lattices of breadth two by Jörg Stephan (Order, June 1993) says this in page two.