What version of ArcGIS are you using? Can you try the Delete Field tool (Data Managment toolbox - Fields toolset)? If that doesn't work, could you share your table?
I downloaded the data and deleted the Text field in the Schools2010.shp without a problem. I used 10.0 SP3.
Can you try restart ArcMap and try deleting the field again? Also you can try the Copy Features tool to make a copy of the shapefile and try to delete the field again. Does that work?
Can you add another field using Add Field tool and then Delete Field to delete that new field?