I've been to approximately 225 games in Tiger Stadium and that was the only time I ever went on the field.
What was funny was how security (25 or so EBR police and a few others in those pre-9/11 days) tried for a few seconds to hold back the flood of students onto the field. Once they gave up, the non-students also started going on the field. After about 5 minutes of that was when I climbed on a folding chair that everyone was using and jumped the fence (cyclone in those days). Security was just standing around by that time.
ETA: Go to around the 11:00 mark of the video linked in the OP. The guy trying to go on the field a few seconds early at the 11:05 mark and getting pushed back
This post was edited on 10/4 at 9:39 am