There is a lot of research into understanding expressivity of first-order logic, or formal languages in general. I am particularly interested in expressivity of graph properties. So for e.g. existence of a $k$-length path can not be expressed in first order logic with two variables. And it is easy to find two graphs which agree on all first order logic formulas in two variables, but differ on paths.
But is this always the case? i.e.:
For given property that is inexpressible in FOL with $k$ variables and $q$ quantifiers, are there always two graphs that agree on all FOL formulas in $k$ variables and $q$ quantifiers, but differ exactly on the inexpressible property?