We present hardness of approximation results for Correlation Clustering with local objectives and for Hierarchical Clustering with dissimilarity information. For the former, we study the local objective of Puleo and Milenkovic (ICML '16) that prioritizes reducing the disagreements at data points that are worst off and for the latter we study the maximization version of Dasgupta's cost function (STOC '16). Our APX hardness results imply that the two problems are hard to approximate within a constant of 4/3 ~ 1.33 (assuming P vs NP) and 9159/9189 ~ 0.9967 (assuming the Unique Games Conjecture) respectively.