Sorry for th late reply on your question, but yes, you can.
If you look at the bottom screen on your DS, you will see the Poke Balls of mine in the top right hand corner. Normal colored ones have no status, grey ones are Pokes with status problems, and black ones are Pokes that have been KO'd. If you count the number of grey Poke Balls before putting a suspected Natural Cure Poke to sleep, and then after it switches out, the number of grey Poke Balls goes to one less on the switch, then you know for sure it had Natural Cure. If this happens, feel free to try and put another Pokemon to sleep.