Consider three distinct atoms of a lattice $a,b,c$. When can we rule out the possibility that $c\le a\lor b$?
So, to be clear, the question is: What is the weakest natural property of a lattice that rules out the above possibility?
It seems like we have this configuration for the standard "diamond" lattice example of a non-distributive lattice, so is this something that can only happen in non-distributive lattices? (This certainly can't happen for lattices with a unique irredundant join of join-irreducible elements representation...)
In a distributive lattice, $c\wedge (a\vee b) = (c\wedge a)\vee (c\wedge b) = 0 \vee 0 = 0\ne c$ if $a,b,c$ are distinct atoms. So $c\not\leq a\vee b$.