I have definitely run into this before ... in fact a 'coffee house' here in RI would have live musicians who wanted to play for tips - and the 'exposure - had to stop doing it because ASCAP/BMI wouldn't let the musicians play any music that wasn't 'original' ...
There is a lot of good debate that can be had about this subject - in fact there has been some right here at S-Z ... it's all about the rights of the writers of the music to receive what is due them ... as I understand it, fees paid to ASCAP and BMI are somehow funneled back to the composers of the music, which only seems fair ... I just wonder how that happens ...