The water came from my neighbor's condo. Must I wait for their insurer to accept responsibility?

Notify your own insurer and the association promptly rather than making repairs depend on a neighbor's response. Document the source and damage, and ask who will coordinate necessary mitigation. Coverage for your property, legal responsibility for the leak and recovery of a deductible are separate questions. For Illinois condos, the statutory insurance provisions and association documents affect master coverage and deductible allocation; the source of water alone is not a complete answer. Ask for the proposed allocation in writing and get state-regulator or legal help if it remains disputed.