If a number of tree limbs from my neighbor's trees fell into my property during a storm, who is responsible for clean-up?

If your neighbor knew or should have known that the trees were sick or that they were likely to lose limbs in a storm--for example, as stated, that they were obviously sick or injured; that had previously shed an excessive number of limbs (or excessively large limbs) during a storm; that the limbs extending over your property were too long and drooping--then he may be responsible. If did not have any warning that these were other than normal, health trees, he would not be liable.

Of course, even if he's liable, if he won't take responsibility, you'd have to sue him, which is almost certainly not worthwhile unless it would be very expensive to clean up.