My high school did almost the same thing. If you failed any of the first 3 marking periods with a grade below 50%, it would be bumped to 50%. For the 4th marking period and the final however, the grade would not be changed.

I don't see an issue with this grade bumping. It prevents one bad marking period from completely screwing a student over. I know it saved my ass a couple of times (English and French...blah....)