Bea first thing, learn from your city if at the time your garage was built if it met zoning code for setbacks if any at the time It was built, THEN learn if your city has any ordinances addressing rain run off from roofs in regard to gutters being needed or not. If you learn that your not required to put up gutters with downleads then thats the end of it. OTHER wise if you know more facts about the nieghbors fence that info will help you, LIKE how old is it ? Was it built from treated or a rot resistant wood? If the fence was built of a cheap softwood like pine and is quite old then your arguement can be the fence is old and you have nothing to do with it. Your neighbor can threaten all they want BUT proving your responsible for the damage is another thing, Do your self a big favor take pictures of the fence now before any parts of it are repaired or replaced, have the place you take the film to date the pics , keep your reciept with the pics. This will give you something to show a court if you must the fences conditon and age.