The K map does not necessarily "fail" for higher dimensions. The problem is that it is so difficult to visualize for more than five variables. A 4 variable K-map is 2 dimensional and easy to visualize. A 5 variable is three dimensional, but is still manageable from a visualization standpoint, because the 2 states of the 5th variable only require visually moving from one plane to the next, without moving in the x or y directions of either plane. Just getting equations correct with more than 5 variables is difficult enough using the K map, much less considering an optimum set of terms ("core" prime implicants and "choice" prime implicates).