We address interactive voice services over Best Effort packet networks where traffic is subject to unpredictable congestions. The quality of voice services can be significantly affected by network delay and jitter due to the rejection of late packets. LIFO has proven to be an interesting overload strategy in delay constrained systems. We propose the use of the LIFO discipline in the context of congested packet networks where voice packets are subject to delay bounds. We analyze how delay accumulates in congested queues and show that serving voice packets in LIFO improves user-level quality. An extensive simulation study shows that LIFO scheduling in congested nodes significantly reduces the fraction of late packets compared to the FIFO discipline. The enhancement of user-level quality is emphasized by using the ITU-T P.861 standard for perceptual evaluations of reconstructed voice. Finally, we show how the proposed mechanism can be easily used in the Internet.