As has been said before, fly fishermen and match anglers can only fish one rod, and their licences allow the use of two.
Nothing quite like having two 20 pound carp in one landing net at the same time, as happened to me once on Crabmill.
On another (easy) water Nr. Sandbach, it's not unusual to have 2 fish on at once.(only 2 rods allowed).
On yet another local water about 2 months ago, I saw an angler playing 2 at once, after a friend helped with 1 fish, his 3'rd rod was away..........jammy git.
I personally like to fish 2 carp rods and use a 3'rd rod for catching roach etc.
I only use 3 carp rods when fishing at night, and no danger of even 2 going off at once as I rarely fish that type of water, and certainly not at night.
Unless you are a long way from any other angler, I think 3 rods is adequate.