That does happen. A lot of times your insurance company will submit a lower offer. What you do next is submit another offer. The offer needs to be higher than their offer, but still lower than your original offer. They are going to come back with a counteroffer to counter your offer. You should be able to get this fixed in a few phone calls. It is called negotiating. You do need to learn the art of negotiating when you deal with insurance companies and claims. They generally do refuse to pay the original offer. They want to get it down to an amount they hope you find satisfying.
What happens when that does not work?
Some companies are going to play hardball. You need to learn how to throw the ball harder, that is all. One thing you can do is ask them to justify why they are offering such a low offer. You will need to write them a "Reservation of rights" letter. You need to get them to be honest. They need to prove why they are doing what they are doing. Some insurance companies are going to try to come up with some flimsy excuse. Do not buy into it. Hold them accountable. They do not want to pay. You need to make them pay.
You can accept their low offer if their reasons are justified. However, it is in your best interest to keep negotiating. Write down everything they tell you. That way you have a record of it. That way they cannot claim "he said, she said" as a justifiable out. You can try to play the emotional card, but that rarely works. Your best bet is to keep everything on file. Submit as much information about what has happened as possible. Keep negotiating. You never accept the first offer. You need to be smarter than they are.
It also helps to talk to another representative at the company to see what they say.