Nash and Milnor were involved in experimental tests of n-person game theory at RAND in 1952. Even then it was clear that game theory had little direct applicability in the real world. See also What Use Is Game Theory? , Iterated Prisoner's Dilemma is an Ultimatum Game, and, e.g., The Econ Con or Venn Diagram for Economics.
A Beautiful Mind: ... For the designers of the experiment[s], however, the results merely cast doubt on the predictive power of game theory and undermined whatever confidence they still had in the subject. Milnor was particularly disillusioned. Though he continued at RAND as a consultant for another decade, he lost interest in mathematical models of social interaction, concluding that they were not likely to evolve to a useful or intellectually satisfying stage in the foreseeable future. The strong assumptions of rationality on which both the work of von Neumann and Nash were constructed struck him as particularly fatal. After Nash won the Nobel Prize in 1994, Milnor wrote an essay on Nash's mathematical work in which he essentially adopted the widespread view among pure mathematicians that Nash's work on game theory was trivial compared with his subsequent work in pure mathematics.For more discussion, including specific experimental results, see Machine Dreams, chapter 6.2: It's a world eat world dog: game theory at RAND.
In the essay, Milnor writes:
As with any theory which constructs a mathematical model for some real-life problem, we must ask how realistic the model is. Does it help us to understand the real world? Does it make predictions which can be tested?...
First let us ask about the realism of the underlying model. The hypothesis is that all of the players are rational, that they understand the precise rules of the game, and that they have complete information about the objectives of all of the other players. Clearly, this is seldom completely true.
One point which should particularly be noticed is the linearity hypothesis in Nash's theorem. This is a direct application of the von Neumann-Morgenstern theory of numerical utility; the claim that it is possible to measure the relative desirability of different possible outcomes by a real-valued function which is linear with respect to probabilities .... My own belief is that this is quite reasonable as a normative theory, but that it may not be realistic as a descriptive theory.
Evidently, Nash's theory was not a finished answer to the problem of understanding competitive situations. In fact, it should be emphasized that no simple mathematical theory can provide a complete answer, since the psychology of the players and the mechanism of their interaction may be crucial to a more precise understanding.