Multi-connectivity facilitates higher throughput, shorter delay, and lower outage probability for a user in a wireless network. Considering these promises, a rationale policy for a network operator would be to implement multi-connectivity for all of its users. In this paper, we investigate whether the promises of multi-connectivity also hold in such a setting where all users of a network are connected through multiple links. In particular, we consider a network where every user connects to its k closest base stations. Using a framework of stochastic geometry and probability theory, we obtain analytic expressions for per-user throughput and outage probability of $k$-connectivity networks under several failure models. In contrast to the conclusions of previous research, our analysis shows that per-user throughput decreases with increasing k. However, multi-connected networks are more resilient against failures than single connected networks as reflected with lower outage probability and lead to higher fairness among the users. Consequently, we conclude that rather than implementing multi-connectivity for all users, a network operator should consider it for its users who would benefit from additional links the most, e.g., cell edge users.