I have had a few experiences with this, students cry at lessons due to many different things.
Sometimes, like others have said, cry for reasons not linked to the lesson itself, something could have triggered it, I remember one of my students had a breakdown during one lesson because I had said to her she needed to spend more time on piano practice prior to her piano exam. She started crying heavely because there was too much pressure at home to do well in maths and english exams coming up, as well as piano and another instrument..So the student felt that no one understood where she's coming from and what it feels like to be a teenager going for her uni exams. I talked to her the whole lesson about these issues and explained I had experienced this not so long ago, and I did have a talk to her parents expressing my concerns.
Sometimes students cry for their own frustrations, frustrated because they cannot get one bit of the music right, or does not understand how to correct something or answer something. So it is best to encourage speaking their concerns rather than keeping it in, for example, one of my students was confused about a cross-rhythm in a piece, I asked her what was troubling her but she kept silent and tears started to swell in her eyes, then 10min of nearly silence, before I realised she didn't know how to practice that bit to get it right, (silly me), so it is important to encourage students to tell you all the troubles they are thinking, not just keeping it inside, I would eventually get it but it would save all the silence and frustration and tears.
I agree with whynot in how important it is for words we choose to use when we're teaching and communicating, there are key words which encourages the student to practice/learn etc and ways of pointing out problems and mistakes without negative connotations