The very common 1761-NET-ENI from Allen-Bradley can support six TCP connections; two outgoing, two incoming, and two either way.
I've never examined whether those TCP connections are also 1:1 with a CIP connection, or if the Net-ENI is always using Unconnected messaging.
In practice, I limit my Net-ENI connections to one HMI and one programming terminal. These devices are notorious for their low TCP connection capacity, and it's easy to accidentally use up TCP connections with RSLinx browsing or other diagnostic functions on the network.