You gotta becareful with that stuff...it usually states the roof, glass and under the car is not covered. Never booked with Budget though so their rules may be different but as far as Dollar, Alamo etc.
I usually book with a UK website to book cars as it's usually 25% cheaper than going thru a Norwegian website and in most cases they say...the roof, glass, tyres are not covered but do real the FAQ's of the website you booked with as it should say if you're covered for that or not.