So can anyone say "I rank at X position for this keyword"? My answer would be no, because things change, searches are relative. But you can point to traffic, number of Facebook followers, Twitter, newsletter subscribers...things like that.
Right. You can point to more of your pages getting indexed, less error codes being reported, more traffic coming from organic search, more traffic coming from links, more of that traffic converting, more pages being read per session, lower bounce rates, more money being made (or whatever your KPI may be). You can look at trends indicating improvement in rankings and higher clickthrough rates.
But if you're opening your browser and running a search in order to prove that you're #3 for some search, you either don't know what you're doing or you're counting on the person you're trying to convince how great you are not knowing what you're doing.