Is the landlord liable for the tenants water damage to a neighbor's unit?

I assume you don't own the downstairs unit. The short answer is yes, in that you are definitely going to be sued for the damage. The longer answer is, it may be a question of notice. If the downstairs neighbor could have alerted you to the leak in time to prevent the damage, you can argue that they contributed to the damage. If the tenant failed to notify you, conceivably you could cross-claim the tenant. Facts will make a difference here. Was the bathroom floor properly done? (If so, why did a single overflow cause leakage to the unit below?) Was the only time the toilet overflowed? Were you on notice that there was a plumbing problem? Etc.