In this short note we provide the examples of pairs of closed, connected Legendrian non-isotopic Legendrian submanifolds (Λ − , Λ + ) of the (4n + 1)-dimensional contact vector space, n > 1, such that there exist Lagrangian concordances from Λ − to Λ + and from Λ + to Λ − . This contradicts anti-symmetry of the Lagrangian concordance relation, and, in particular, implies that Lagrangian concordances with connected Legendrian ends do not define a partial order in high dimensions. In addition, we explain how to get the same result for the relation given by exact Lagrangian cobordisms with connected Legendrian ends in the (2n + 1)-dimensional contact vector space, n > 1.