I know Wikipedia isn't a great source, but in this case, Wikipedia seems to have good references, so I'm going to quote Wikipedia because it's concise.
John Nash never worked for the Pentagon, so he never would have had the chance to do that, as portrayed in the film.
Wikipedia says this:
...in real life Nash spent his years between Princeton and MIT as a consultant for the RAND Corporation in California, in the film he is portrayed as having worked for the Pentagon instead. (Source: A Beautiful Mind DVD Commentary)
Further information about Nash's career can be found on the wikipedia page specifically about John Forbes Nash, Jr.