There are no problems when I open the file on my ArcMap. I can edit and delete the Score field:
Some possible solutions:
1. Try to delete/change the field on the rar file you've sent me. If it works then your original file might be corrupted.
2. Did you join fields before you tried to delete/change? If yes this might have to do with a joined field not responding to your editing because it was actually not a part of the feature class.
3. Try to open the feature class in a new project and test.
4. Are you reading it from an SDE? If yes try to export to local and then edit. If it works then it's a lock issue on your SDE.