In this talk, I will discuss quantitative questions about Lagrangian cobordisms between Legendrian submanifolds. In particular, given two Legendrian submanifolds, how long must a Lagrangian cobordism that joins them be (if it exists at all)? This can be viewed either as a symplectic non-squeezing phenomenon for Lagrangians or as a result about the displacement energy of a Lagrangian cobordism. I will outline the tools necessary to prove non- squeezing results that arise from a filtered version of Legendrian contact homology, a Floer-type invariant. This is joint work with Lisa Traynor.