Can Parakeets commit suicide???

So my mother has birds. 4 parakeets to be exact. And she loves them. A lot. She has already lost 2 and is can't stand the thought of losing another. We were talking tonight and she was telling me that one of the birds keeps dropping off her perch onto the floor, and once even got her self hung up in some rope. She is telling me this and laughing about what a clown this bird is, and I am thinking "OMG it's trying to kill itself." I mean c'mon if I lived in a small cage with 3 other things and a lady kept trying to pet me, yeah I think I'd jump. I am just hoping this little jumper waits a while before she dies.