Microservice applications rely heavily on DNS for internal service discovery in ephemeral environments. Unlike traditional public DNS, cloud-native DNS operates as a core infrastructure component with strict requirements for correctness, low latency, and adaptation to frequent record changes. There is a gap between existing DNS research focused mainly on performance in public environments and the unexplored resilience of internal, cloud-native DNS systems, especially those supporting private zones in containerized environments.This paper addresses this gap by examining the role of private DNS in cloud-native systems and applying resilience testing techniques to systematically evaluate its behavior. First, this work analyzes public cloud incident post-mortems and finds that DNS is rarely the primary root cause of outages, and incidents disproportionately affect private DNS over public DNS. Secondly, we design and evaluate a resilience testing strategy for a multi-cluster cloud-native DNS system published by Swisscom AG, the biggest telecommunications provider in Switzerland. The proposed strategy uses chaos injection and DNS traffic generation to evaluate availability, latency, accuracy, and recovery behavior under different disruption scenarios. We find that the system operates within acceptable limits in most scenarios and resilience testing can reliably capture the impact of caching, resolution inconsistencies, and failure-handling weaknesses across test cases.