Cold start car always starts. After running the vehicle for a while (5 minutes or more), shut off the car for a few minutes, then car will not start. All dash lights illuminate and general power seems fine, but vehicle doesn’t crank…has the same symptoms as a dead battery. Let car sit for hours (cool down), then it will start. Typically, the starter does sound a bit rough when the car does turn over. This was an intermittent problem in the past, but the frequency has increased to where it now happens almost every time the car is used.
I also have a '97 Maxima (123k miles). In addition to the starter as an obvious possibility, please also consider the ignition switch. In Sept of this year I had to have the switch replaced on my car, and the Nissan tech who did the work said that when these switches start to wear out they can lead to problems like your car is having. It took the tech about 20 minutes to do the replace it – problem solved.