Apartment to apartment damage in a condo

Depends what risks your insurance policy covers

NYOWNER:
First, you need to determine if you actually caused the damage to "another" unit.
By "another" do you mean the unit downstairs from yours?
Did you cause a leak, either by an overflowing tub or sink? Was it because a reno was done and fixtures were not installed properly?

It may be the Condo's fault ( if pipes are leaking ).

You need to know the origin of the "damage". Either it's your fault -- or it's not.
You would know.

If it's your fault, arrange a payment agreement through the managing agent.
You could then resubmit your insurance claim (no insurance company will approve a claim that is not
proven to be your fault: "we may have caused damage..." isn't sufficient).

Let them sue you and provide evidence. Not sure why you are volunteering liability. What you need is a good dose of yikes, that'll show your neighbor.

huntersburg:
That's a good way to go, only if the Condo's super investigated and reported that NYOWNER "may have caused damage to another unit..."

NYOWNER , if knowing that there was no water overflow or reno problem, can get an honest plumber to inspect NYOWNER's apartment to verify that the leak (or whatever the "damage" is) did not originate in NYOWNER's unit.

In any case, NYOWNER will need such proof. Of course, "never volunteer liability" unless the unit owner has shown good faith in having an inspection done (NOT only by the condo's super, or even "A plumber, sent by the managing agent) and the "damage" is found to be caused by NYOWNER.
The reason for this is, that if NYOWNER is responsible for the "damage", such "damage" can spread and get worse, while the "other unit" goes through the process of litigation.

NYOWNER can contact Primer, who will recommend an honest plumber, to inspect and provide a comprehensive report.

There is no such thing as "a good dose of yikes".

The other unit owners should submit a claim to their insurance company for the damage.

Their insurance company will assess, pay the claim and then look to your insurance company for reimbursement, if they deem the damage is your fault.

This is called subrogation. Even the injured party's deductible is likely to be recovered.

This probably sounds scary for you but actually, I had an $80,000 damage claim that played out this way (I was the victim), and the insurance companies basically battle it out behind the scenes.

What your condo bylaws do require, we hope, is insurance. This is why.

Flutistic:
In theory, that should be the case.
In reality, the damaged unit could be owned by the Sponsor or a Condo Board-member.
The super is called by them to go to NYOWNER's unit and say that the damage "may be" caused by
NYOWNER.
Otherwise, the super would have inspected the damage and reported back to the management company. The management company would have contacted NYOWNER to verify that NYOWNER caused the damage, with a detailed report of when/how the damage occurred and photos of the damage.
NYOWNER would have been given evidence of the damage he/she caused ( or, what the Condo Board
and Management Company claimed was caused by NYOWNER.) NYOWNER would have that, to include in his/her insurance claim.

That the other Unit owner did not submit a claim, sounds not Kosher.