sethmopod
I think part of my thing on the 20 times bit comes from being classically trained.  I know there's a thing among a lot of pop/rockers (which I am at heart, despite two classical music degrees) that the first take is usually the best take.  While I have had that happen to people I've recorded, it has been extremely rare.  I find that I get much better results from musicians when I actively coach them through their takes.  This is true for singers and instrumentalists, and I've worked with some really good ones. 
 
It helps in this that I am a sought after and successful teacher in my area; I'm used to improving performances without smashing egos.  I've studied with some of the best teachers out there (Larry Hurst recently retired from Indiana University - if I can grow to be half the teacher he was, I will count myself as a crazy success), and one of the great lessons I've learned is that energy in music can and must be planned.  Any less is the sign of a hack. (<-- this is a harsh statement and probably overdone in the spirit of an internet rant)
 
A very common process I find when working with singers is that we take a couple of full passes at the song to get warmed up.  I may make a few general comments at this point, but usually nothing too much.  Then we starting going through the song phrase by phrase examining the articulation and dynamic structure of each one in much greater detail.  I find that most singers have not done this - they're used to giving run-throughs.  Any good music teacher can demonstrate readily that this is ineffective practice and misses a lot of the detail that leads to a great performance - in any style.
 
By the time we work through the song to the end, I usually find that the singer has found a new groove that manipulates the elements of musicality a lot more effectively.  They are usually singing in a slightly different style by the end of the song than they were from the beginning.  At this point, we will usually go back and redo earlier verses and choruses to match the final style.  What results has been, in my experience, always superior to what those first couple of takes were.  With a reasonably good singer this whole process takes about an hour or so.  Bear in mind, this is not an hour of constant singing - voices can't stand up to that.  It includes discussion, analysis and some rehearsal.  There are also a lot of takes built up.  The editing usually isn't bad, though, because the keeper is almost always immediately apparent - they hit that interval that was giving them trouble, they lock in with the groove, the articulations and accents are there to give it energy, etc.
 
I've had it happen where no matter what I've tried it just hasn't clicked.  I had a session recently where I had to send the singer home to practice.  He took it well, though, because I was helping him understand why his takes weren't getting what he wanted.  He came back a couple of weeks later and nailed it.
 
A caveat here on the "do it again until it's right" comment that started all of this.  Just doing it over and over without the coaching to improve it each time is a waste of time and will leave everyone frustrated and defeated.
 
I'm not usually given to internet rants like this, but I guess I needed to get it out.  I'll shut up now.
 
Seth