I remember my uncle was not allowed to rent a car at all for multiple reasons

1. he declared bankruptcy and couldn't get a credit card, and obviously the soft pull Hertz pulls wouldn't let him rent with a debit card
2. His driving record was atrocious. He had more speeding/running red light/stop sign violations than I had fingers... At one point, he couldn't even get auto insurance

People like my uncle are out of the ordinary obviously, but I doubt a speeding ticket here and there and a fender bender once in a while will affect your ability to rent.. a DUI on the other hand...