It's certainly possible. No doubt about it. I've checked it on myself and still continue to improve.
I would even say if you really want to learn and are ready to put the time and effort in it -- you are going to succeed even on your own. But without an effort - the won't be any results.
And if you are lazy -- no matter what school or tutor you have -- the results will be poor.
Considering the abundance of learning materials and multimedia, you can be very effective on your own. I think video lessons, podcasts etc. can substitute a teacher almost completely.
The only thing you need is to be persistent and methodical.