WHILE ENROUTE FROM GRAND ISLAND, NE TO NASHVILLE, TN, THE PILOT NOTICED THAT THE AIRCRAFT WOULD NOT TRIM NOSE DOWN, BUT THAT IT WOULD TRIM NOSE UP NORMALLY. THE PILOT REDUCED SPEED AND ALTITUDE AND DIVERTED TO ST. LOUIS, MO IN ORDER TO HAVE THE PROBLEM REPAIRED. ON SHORT FINAL TO RUNWAY 30R, THE AIRCRAFT PITCHED DOWN VIOLENTLY. THE PILOT STATED THAT BOTH PILOTS WERE REQUIRED ON THE ELEVATOR TO EFFECT RECOVERY FROM THE PITCHOVER. ONCE RECOVERED, THE APPROACH WAS CONTINUED TO A NORMAL LANDING. INVESTIGATION REVEALED THAT ONE OF 4 ELECTRICAL TRAVEL LIMIT SWITCHES HAD A LOOSE CONNECTION WHICH WOULD ALLOW THE ACTUATOR TO DRIVE TO THE MECHANICAL STOP. AFTER NUMEROUS CYCLES, THE ACTUATOR SLEEVE/DUST COVER WOULD CRACK ALLOWING THE ROD ENDS TO DISCENGAGE FROM THE ACTUATOR. IT WAS ALSO DETERMINED THAT WITH THE MECHANICAL FLEXIBLE SHAFT INTERCONNECT BETWEEN THE TWO ACTUATORS A FAILURE OF ONE ELECTRICAL LIMIT SWITCH WOULD PERMIT BOTH ACTUATORS TO BE DRIVEN TO THEIR MECHANICAL LIMITS.