Florida law recognizes a concept called comparative negligence. Comparative negligence is how insurance companies and juries apportion (or distribute) fault. So if you are entitled to $10,000 for injuries sustained in a car accident, but you are found to be 50% comparatively negligent (i.e. that the accident was 50% your fault), you would be entitled to only $5,000.
Insurance companies make it their mission to try to find any reason to assign you blame in an accident. This is just one of many reasons why you need a personal-injury attorney to help you fight this fight.
I have seen situations where police officers, who I have the utmost respect for, make mistakes when they issue a ticket or document an accident. Just because the police officer gives you a ticket or citation, does not mean he/she properly investigated the facts (i.e. listened to your side of the story, elicited the germane facts from witnesses, etc.). Remember, the officer tends to arrive at the scene of an motor vehicle accident AFTER the accident occurs! So they are not always in the best position to figure out who is really at fault.
Lets suppose you run through an intersection, as a light turns from yellow to red, and hit another car. If we can show that the other car was speeding, failed to properly watch out for oncoming traffic, etc…, the accident may be deemed mostly your fault, but I would argue that some percentage should be properly assigned to the other driver for his failure to drive carefully and safely.
Obviously every situation is different. But when an accident is mostly your fault, a recovery may still be possible (i.e. 20% of something is better than nothing). Obviously, these are the more challenging cases, but certainly worth investigating.