You should never have more than one DHCP server on your network (assuming they are on the same subnetwork). That will cause problems, but I'm not sure about your specific issue. I would shut the Nest routers DHCP server off.
[Be nice! We are mostly fellow Verizon customers. My setup: 75/75 Mbps; Linux router (2x 1Gbps ports) + iptables; Netgear Orbi x 2