You may be thinking of number of ports. There are 65536 ports available in the TCP in the current versions of IPv4. This is not just a Linux limitation, its part of the protocol. Your IP address identifies your machine, and the port identifies a program on your machine.
But, the number of connections isn’t limited by that. A connection consists of 5 pieces of info, in geek speak a 5-tuple. It is determined by protocol (TCP, UDP), local IP address and port, and remote IP address and port. So, take a webserver. It can service many connections on the same port (most likely 80). Your webserver can even support multiple connections to the same client machine. Say, you’re connecting to google.com from two windows. Your machine will pick an unused port for each connection. So, google’s server will have to keep track of (TCP, google.com, 80, yourmachine, someport1) and (TCP, google.com, 80, yourmachine, someport2). At some point you’d bump into limits, but it’s not a hard limit, and is very system dependent.
And yes, each socket is a file descriptor, but not all machines use shorts for the fd table. On my system, a not tuned at all system,
cat /proc/sys/fs/file-max gives 323997. I’m sure I could boost it if I needed to.
So, there is a limit of 65336, but it has to do with addressing, not number of connections. Number of connections is limited, but more by system config, and how much memory it has.