You can make a claim, but it doesn’t guarantee you’ll be compensated by your neighbor’s home insurance policy for the damage to your shed.
If there is no law in your area regarding bamboo, the insurance company may easily say it’s not responsible for the bamboo that has crept into your yard and that you should take care of it yourself, which is your right once it is on your property.
Or, if your neighbor just moved in and the bamboo was planted by the previous owner, then his insurance company could say he was not negligent because did not plant the bamboo and thus not liable for your damages.
However, if your neighbor did plant the bamboo and there are laws or ordinances in your area against this, then it is likely your neighbor’s insurance company could find him negligent and accept your claim.
If his home insurance provider doesn’t take your claim, you could speak to your homeowners insurance company and see if the damage to your shed is covered under the portion of your policy that covers “other structures.”
If you run into dead ends with home insurance claims you could take your neighbor to court in hopes of obtaining a judgment against him for the damages and removal of the bamboo. Some states allow this under a nuisance law.
For example, Connecticut’s Gov. recently signed into law Public Act 14-100 that profits people from planting invasive bamboo (here it’s called “running bamboo”) on their property within 40 feet of abutting property or public right of way. The law allows a violator to be subject to a $100 fine – and each day of a continuing violation is a separate offense so the fine amount can add up fast.
Connecticut law also declares that running bamboo that grows beyond a person’s property line is a nuisance (a condition that interferes with the use or enjoyment of one’s property) and is a civil issue for which one can sue for damages, including the cost of removing bamboo that spread to a neighbor’s property.
I’d recommend you check into your local and state laws to know where you stand, and then go forward trying to make a claim with your neighbor’s home insurance.