Looking at the RFC for TCP: RFC 793 - Transmission Control Protocol, the answer would seem to be no because of the fact that a TCP header is limited to 16-bits for the source/destination port field.
Does IPv6 improve things?
No. Even though IPv6 will give us a much larger IP address space, 32-bit vs. 128-bits, it makes no attempt to improve the TCP packet limitation of 16-bits for the port numbers. Interestingly the RFC for IPv6: Internet Protocol, Version 6 (IPv6) Specification, the IP field needed to be expanded.
When TCP runs over IPv6, the method used to compute the checksum is changed, as per RFC 2460:
Any transport or other upper-layer protocol that includes the addresses from the IP header in its checksum computation must be modified for use over IPv6, to include the 128-bit IPv6 addresses instead of 32-bit IPv4 addresses.
So how can you get more ports?
One approach would be to stack additional IP addresses using more interfaces. If your system has multiple NICs this is easier, but even with just a single NIC, one can make use of virtual interfaces (aka. aliases) to allocate more IPs if needed.
NOTE: Using aliases have been supplanted by
iproute2 which you can use to stack IP addresses on a single interface (i.e.
$ sudo ip link set eth0 up
$ sudo ip addr add 192.0.2.1/24 dev eth0
$ sudo ip addr add 192.0.2.2/24 dev eth0
$ ip addr show dev eth0
2: eth0: <NO-CARRIER,BROADCAST,MULTICAST,UP> mtu 1500 qdisc
pfifo_fast state DOWN qlen 1000
link/ether 00:d0:b7:2d:ce:cf brd ff:ff:ff:ff:ff:ff
inet 192.0.2.1/24 brd 192.0.2.255 scope global eth1
inet 192.0.2.2/24 scope global secondary eth1
Source: iproute2: Life after ifconfig