I know there are some Trek fans on here and I just came across this story a few days ago even though the short is over a year old.
Long story short is in late 2014 there was a crowd funded fan made short-film (20 mins) made called "Prelude to Axanar"
https://www.youtube.com/watch?v=1W1_8IV8uhA
Before I go on, the short in and of itself is awesome. If you can, I suggest you put it on your TV in HD full screen and turn it up etc. It takes place post Enterprise era (Backula) but pre Original Series and is in documentary format about a four year war between the Federation and Klingon Empire.
Now, "fan film" has become a subjective term, because some of the recent Star Trek projects made by "fans" are extremely professional and this is obviously a high end production with former Trek actors etc.
Paramount and CBS have always been lenient of fan projects and in some cases have encouraged them. There have been many big budget projects allowed to proceed with the stipulation that they remain not-for-profit.
As mentioned this short was posted on Youtube in late 2014 and caught fire among fans. Apparently (after it came out) via kickstarter etc, over a million dollars was raised by fans anxious to see this made into a full length feature.
Then, just a few months ago in late December 2015, Paramount/CBS filed suit for alleged copyright infringement.
http://variety.com/2015/film/news/st...ed-1201669522/
Seems like quite the story brewing.
Anyway, if you haven't seen it, check it out at the above link. Definitely some very cool stuff imo if you like Star Trek and it will be interesting to see where this ends up.