I love how clean cats are. Even when they occasionally have an accident, they do their best to try to fix it. Nahum sometimes misses the litter box but he always makes sure he still covers up his business. If it occurs on a rug, then he'll fold the rug over the accident.
As a kid, I remember that our Chihuahuas would have frequently have accidents on the carpet. Their approach was to try to hide it in an inconspicuous spot (under a table, or behind the couch).