Oh yes - I have done this many times in my MS time and I've come close several times during my 2 years so far for my PhD. I think they are mostly tears of frustration that something isn't going the way I want/expect but sometimes it's tears of other stuff (sadness, failure, whatever).
My three tips - 1, bring water with you to any meeting - you can sip and get a break before you bawl. 2, bring tissues with you, just use them discretely as needed. 3, look at the ceiling, floor, wall, whatever to get away from the piercing eyes (I find the ceiling best, as tilting my head up reduces tears falling out).
I find it makes most supervisors uncomfortable, so I often apologize as soon as it starts and ask if I can meet again later after I regroup. I also let them know it's not THEM, but its the derailing of the plan/whatever...
But yeah, people reach a breaking point and then crack and research can trigger that very quickly, just move on from it.