This is definitely not true. Anyone who hoops knows that when you get tired, your footwork gets sloppy, you get less lift, and your stance becomes narrower. All of these things increase the chance of rolling your ankle.
In Rose's case it may or may not have been related to fatigue. Theres no way for us to know, especially without seeing the specific play. When a guy gets hurt in the 1st game back after missing serious time, people have to question the coach's decision to give him 40 minutes and the training staff's decision to clear him.