Firstly, I should say that I am not giving you formal legal advice and in any event you must and should not rely on any views or information contained in this post in any subsequent proceedings of any nature or for any other purpose.
I am quite sure you are more than sensible enough not to rely on some guy from the internet anyway(!) and will be aware that you should seek formal legal advice whenever faced with any kind of claim.
However, some general points that might be worth considering.
Now, that seems like an odd thing to do. There was no negligence and it was clearly an accident. Surely it's just a matter for the insurance and I couldn't be personally liable for the damage. If my bath fell through the ceiling they wouldn't personally bill me for thousands of pounds to fix the ceiling, they would just claim on the building's insurance.
1. Negligence or breach of duty can arise by act or omission and anyone living in a block of flats will more than likely have a duty to keep in repair and maintain installations such as the plumbing for the bathroom and, arising from this, a potential liability to compensate a third party injured by a negligent act or omission (for instance failing to keep in good repair and maintain plumbing). A contract between the owner/landlord and the management company (or similar) would likely contain more detail on any contractual duty.
2. Generally, the owner of the freehold of the building (or their agents) will have taken out buildings insurance (albeit paid for by the tenants) for the building which would more than likely cover any structural damage etc. However it does not always cover items such as damaged fittings such as ruined carpets and a landlord or owner might be liable for these. The buildings insurance policy would set out the purported areas and level of cover and provide more information.
3. If person A's flat were damaged by something arising from Person B's flat, he might well ask Person B to pay for the repair etc. (this is, of course, not the same thing as bringing formal court proceedings - i.e. suing) even on a speculative basis rather than claim on a policy to which he himself contributes and whose premiums might rise!
Person A can write to Person B asking for anything they want (within reason!) but cannot force Person B to pay anything without a court order to which, in the usual run of things, they would need to prove they were entitled.