Even if the question has an accepted answer I would like to leave my answer to tell you about my encouraging experience with the saxophone.
I never had any musical education but that didn't stop me from buying a Xaphoon In July 2006. I played the thing by ear for a few months. It was dreadful at start but the sound got better sounding after 6 months. Around this time (Feb 2007) I met a sax player at a party and asked him if he would give me some classes which he did; and after trying a Tenor and a Alto sax I bought myself a Alto saxophone: a beautiful copper coloured Jupiter Colorado (I don't mean to publicize it but I'm so proud of it still today that I cannot resist... :-) Note that at this point I still had no musical education but had some money to spend on it and definitely a huge willingness to learn music and how to play it.
A couple of months after starting these classes I realized that to play by hearing it, would require much more time practicing than I could afford so I decided to join a music academy where I started learning music and the instrument. This was September 2007.
Three years later (August 2010) I was doing this, and 2 more years on there was some improvement... :-) The video is the same but the sound track was replaced by a more recent one and I guess you can tell the difference (also the recording equipment was much better). Today, two years after that last recording I feel I have progressed even further and all of it was due to a great, great teacher (see Tim's answer ;-). I have also started composing a few pieces and harmonizing them.
Oh, and in 2014 I'm 43 so I started with 36 and to me being 22 is probably what you refer to as "school days"!!! ;-)