Examples of behavioural breakdowns in game theory?
Here is an article that reports about several laboratory experiments showing large inconsistencies between theoretical predictions and observed behavior for some payoff structures: 

Goeree, J.K. and Holt, C.A., 2001. Ten little treasures of game theory and ten intuitive contradictions. American Economic Review, 91, 1402-1422.
https://doi.org/10.1257/aer.91.5.1402

There are also papers studying variants of the ultimatum game in order to quantify players’ preferences for fairness. One approach to do so is to compare (i) games in which two human players interact with each other and (ii) games in which a human player interacts with a computer (so there is no scope for social preferences): 

Hoppe, E.I. and Schmitz, P.W., 2013. Contracting under incomplete information and social preferences: An experimental study. Review of Economic Studies, 80, 1516-1544.
https://doi.org/10.1093/restud/rdt010