Sounds weird! Being half full shouldn't be an issue, it should run fine all the way to empty.
I would double check there is no hop gunk stuck in the dip tube and in the top of the beer post (what screws in over the top of the dip tube).
Also check that your beer line connection to the beer post isn't damaged, because it might not be depressing that little valve enough to get flow.
Then check for kinks in your line beer, and taking apart and cleaning your facet thoroughly in case there is some old gunk cemented in there.
Last thing would be to double check your regulator. It might not be putting the amount of gas into your keg that the dial says. Not sure how to actually check this, but if its reading 20 and actually only pushing 3psi you might have your culprit.
Other than that, I'm as stumped at you are!
A little irreverence is good for the soul