Your chances depend heavily on what condition that tree was actually in before it fell. Most jurisdictions won't hold someone liable just because a tree came down in a storm - that's considered an act of nature and you're stuck with the bill. But the real question is whether your neighbor had reason to know the tree was hazardous. If there were visible signs of disease, dead branches hanging over your property, or if the tree had been leaning precariously, that changes things. You'd need evidence that the tree posed an obvious danger that a reasonable person should have noticed.
Document everything right now if you haven't already - take photos of the damage and what's left of the tree, including any signs of rot, disease, or structural issues. Check if your neighbor ever had an arborist look at it, or if you have any communications where he mentioned concerns about the tree. Some homeowner's insurance policies will cover tree damage from storms without assigning fault, so check yours first. If you want to pursue this further, you might consult a local property attorney who can tell you what your specific jurisdiction considers "negligence" when it comes to tree liability - the standards vary quite a bit depending on where you live.