Jack-knifing happens to lorries when the rear wheels of the towing vehicle lock up and the inertia of the trailer pushes the tractor unit from behind, forcing it sideways and around. A bicycle, being a single track vehicle (and assuming that there is enough weight in the trailer to overcome the weight of the rider) would tip over at approaching 90 degrees rather than jack-knife. A tricycle could be made to jack-knife given enough trailer weight and good enough rear brakes.
I used to tow a Cannondale Bugger (yes Bugger) which attached to the seat post. It wasnt unusual to have the weight of my drunken friend push the back wheel of my bike up in the air under hard braking, a kind of vertical jack-knife if you will.