I agree with the other moms that it varies for each child. However, kids need sleep and sometimes it just takes some adjusting. When my oldest was about 3, he started resisting naps with all his might. So, I moved naptime from 12:30 to 2:00 and he started taking them again without problems. That's not always possible when older ones are in school, but you can play around with times to see what works best.
Also, I agree with others that even if the don't nap, they should have quiet/rest time. You can provide books and toys, and let them know they have to stay in their room. It will be hard at first, but they will come to enjoy this time. A friend gave me a great tip that really works. She tells her kids that they need to lay in bed and read for 30 minutes, and that she'll set a timer. She tells them that she'll check on them when the timer goes off, and if they're awake they can get up, but if they're asleep, she'll let them sleep. I've used this on my 4 and 5 year olds when they really need a nap, but don't want to take one, and it really works well.