I'm convinced that my two young children do not require loads of sleep to function. My 3 year old is consistently getting through her days without ever falling asleep at "nap" time, and it has been months (at least 6) since my 11 month old has dozed off in the car. In fact, that same 11 month old has dropped down to one nap a day already!
Many kids at age 3 are starting to show signs that they are no longer in need of a mid-day nap. My daughter started showing those signs at 18 months. Instead of taking her nap, she would be upstairs playing for an hour and a half in her crib. Near dinner time, she never had a "fussy" period indicating that she needed some sleep a few hours prior. She was also falling asleep easily at bedtime, and sleeping a full 12 hours until the next day. Generally speaking, she gets all of her sleep that a child her age needs at bedtime, and functions all day without needing any nap.
At age 3, she still rarely falls asleep during "nap" time, but each day, there is a rest time scheduled in. Resting during the day for young children is important for several reasons. Young bodies need time to rest and recover from the day's activities, and resting helps the body grow and maintain healthiness. A young child's body and mind is still growing at a rapid rate in the first 5 years of life, and periods of non-stimulation help the brain to process what it is learning and taking in.
There are days that we are on the go running errands, heading to doctor's appointments or any host of other reasons that would preclude my 3 year old from a nap. Thankfully, she can handle this and doesn't start melting down by 2:00. Yet, I've noticed that on the days that she does not have any rest time due to any of the above reasons, she is more apt to whine in the evenings, be more sensitive, and show signs of crankiness that are otherwise not there. Though she is not a napper any longer, she still needs her rest time like all young children do, and this makes a world of difference in their attitudes and in their ability to focus and learn.
While are days are busy, we need to make their time of rest just as much a priority as we do getting to the bank, the grocery store, and the dentist all by noon.