Question

why dimensional methods are applicable only up to three quantities

Answer 1

because the mass, length,time are the fixed and fundamental quantities...they didn't depend on other quantities and the all force and moments are mainly based on this

Answer 2

For mostly all the physical quantities there are three dimensions involved. They are expressed in the form of mass M, length L, and time T.

Almost all the physical quantities can be expressed in these three dimensions itself. Also when three dimensions are involved only three equations shall be enough to solve.

If the number of unknown quantities increases it becomes a bit hard to solve.