The development of IPv6 stacks for wireless constrained devices that have limited hardware resources has paved the way for many new areas of applications and protocols. The Constrained Application Protocol (CoAP) has been designed by the IETF to enable the manipulation of resources for constrained devices that are capable of connecting to the Internet. Due to the limited radio channel capacities and hardware resources, congestion is a common phenomenon in networks of constrained devices. CoAP implements a basic congestion control mechanism for the transmission of reliable messages. Alternative CoAP congestion control approaches are a recent topic of interest in the IETF CoRE Working Group. New Internet-Drafts discuss the limitations of the default congestion control mechanisms and propose alternative ones, yet, there have been no studies in the literature that compare the original approach to the alternative ones. In this paper, we target this crucial study and perform evaluations that show how the default and alternative congestion control mechanisms compare to each other. We use the Cooja simulation environment, which is part of the Contiki development toolset, to simulate CoAP within a complete protocol stack that uses IETF protocols for constrained networks. Through simulations of different network topologies and varying traffic loads, we demonstrate how the advanced mechanisms proposed in the drafts perform relative to the basic congestion control mechanism.