In September last year I had the misfortune to have a runaway trim in an EMB110 on takeoff at Bankstown. The rotation was normal and as the aircraft accellerated it started nosing over uncontrollably. Application of nose up electric trim did nothing, manually trying to move the trim was unsuccessful. At the time we diagnosed a failure of the trim system through the screw jack or some other component ( the trim appeared stuck).
The only way to maintain some control was to reduce power and airspeed to a point that subsequently allowed both crew to overcome the aerodynamic forces. (FO with feet on the instrument panel pulling on the column for all he was worth - Nice work on his part if I do say so!)
The initial findings were ones of the Captains trim switch sticking irrespective of selected position and the elevator trim torque motor failing a breakout test thus preventing the crew from overpowering the electric trim.
The aircraft had no switching to enable the turning off of the electric trim system other than pulling CB's.
So to answer the question, yes trim runaways can occur, they are a serious event and not always that easy to detect. If your in straight and level flight and the nose starts to pitch down or up then its generally easy to diagnose, but in a state of aircraft acceleration it can be difficult to determine what is a changing aerodynamic force associated with speed to one of a trim change.
Not an experience I care to repeat!