States have negligence laws that help insurance companies be able to determine and divide fault in an accident. There are 2 main types of negligence laws, one is contributory and the other is comparative. North Carolina applies contributory negligence to auto accidents. This means that if you are even slightly at fault in a collision you are not entitled to any recovery against the other driver.
Comparative negligence is different in that if you are found a percentage at fault then that comes off of what recovery you can make against the other party. For instance in Ohio they have comparative negligence laws that state parties share the cost of damages from an accident in proportion to the share of negligence. A motorist determined to be 50% or less at fault may recover their damages minus the percent cause by their own negligence. If more than 50% there would be no recovery for losses from the other party.
Negligence typically comes into play when it is contended that two or more parties failed to perform at the standard of the "ordinary reasonable person". For example, suppose one person was driving too fast in a patch of dense fog on the highway and hit a car but the car that was hit did not have its lights on as it should have. In this case both parties could be held negligence in some way and thus partially responsible.
As the North Carolina Department of Insurance (NCDOI) notes, the insurance adjuster investigating the accident will attempt to determine who is negligent or at-fault. The Automobile Liability coverage will pay for bodily injury and/or property damage for which any insured becomes legally responsible due to negligence. The claims adjuster assigned to your claim should be able to tell you what is included in their investigation to determine fault.
The NCDOI glossary defines negligence as the failure to act as a reasonably prudent person would have acted under similar circumstances.
So yes, the North Carolina Contributory Negligence Law does support what the other party's insurance company has told you. This NC law bars a driver from collecting damages if determined to be partially at fault. In essence, if you contribute to an accident, you may not be able to collect on a liability claim.
Any disagreement you have with the other party's insurer regarding their determination of fault would ultimately have to be resolved in a court of law if you cannot come to an agreement with them on your own. So if they somehow find you 1% at fault and you do not feel you were at fault in any way (and this is just a way to get out of paying your claim) you would likely need to take your claim to court.