If your neighbor’s tree is blown over by storm lands on your property, is the resultanting damage covered by your neighbor’s insurance policy or your own insurance policy?
Most people would assume that the damage is covered by their own insurance policy because the event was an act of God. However, while many policies cover an "act of god", some do not. Some policies even have hurricane damage exclusion clauses and/or deductables burried in the fine print. If you believe that you should have been covered, but were not, it is most certaily time to speak with attorney.
That, said, can you sue the neighbor for damages?
Certainly, hurricanes and other meteorological conditions which result in strong winds are forces of nature qualifying as acts of God. However, simply because a force of nature, or act of God, was a cause of the damage does not automatically mean that the act of God defense applies. With an "act of god" you can't sue, as the neighbor was not responsible for the tree falling.
The biggest issue is whether they knew in advance that the tree or large branches could easily be blown down by wind. If they knew that the tree is dead, for example, or knew that it was diseased or rotting, that could be used as evidence that they are liable for damages caused by the tree.
This is why it is important to communicate with neighbors regarding potential hazards to your property, posed by their property (a dying tree, a rotten shed leaning towards your fence, etc.) in writing or email. So long as you don’t trespass, you can even trim branches that are over your property, and have a neighbor’s tree visually inspected.
Good luck in your recovery!