Question

why dimensional methods are applicable only to three quantities?

Answer 1

if a quantity depends on more than three factors having dimensional formula cannot be derived
Because on equating the powers of M,L,&T on either side of the dimensional equation,three equations can be obtained,from which only three unknown dimensions can be calculated

Answer 2

Now almost all the physical quantities we encounter in engineering and fluid dynamics (where this is mostly used), are combinations of just three of the fundamental dimensions, mass (M), length(L), and time(T).