Question

Why dimensional methods are applicable only up to three quantities?

Answer 1

Before answering this questions , Let see dimension of most important physical quantities. See attachment, here dimension of all important quantities are considered just mass (M) , length (L) and time (T) .
So, according to me , we use three quantities (M,L & T) because these are enough to define dimension of almost all the physical quantities we encounter in engineering and fluid dynamics (where this is mostly used).

Also taking only three quantities is easy to solve equations .
Because to find three unknown , three equations are enough . While when we use more quantities then number of unknowns will be increase , therefore tough to solve them.

Answer 2

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