In this letter, we analyze the downlink performance of ultra-dense networks (UDNs) considering a multi-piece path loss model with line-of-sight (LoS) and non-LoS (NLoS) transmissions, a general multi-path fading, three dimensional transmission distances and a finite user equipment (UE) density. A constant capacity scaling law is obtained, that is, both the coverage probability and the area spectral efficiency approach a constant when the base station (BS) density grows extremely large. However, this performance limit is not always the maximum achievable performance, which means that network densification alone is not always the best strategy. Moreover, the performance difference of various multi-path fading models is more conspicuous with the growth of BS density, and highly depends on the UE density.