If you are willing to make "niceness" assumptions, assuming that the n-type is, say, nilpotent with finite fundamental group, then *the dimension, in any reasonable sense, will always be infinite*, since it will have homology in infinitely many dimensions.

The first result along these lines is due to Serre:

J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–Mac Lane, Comment. Math. Helv. 27 (1953), 198–232.

I wrote a couple of papers about this back in the days:

http://www.math.ku.dk/~jg/papers/postnikov.pdf

http://www.math.ku.dk/~jg/papers/serre.pdf

where I prove that the cohomology, also as a ring, will be huge.

If you don't want to make assumptions on the fundamental group, then it is an open problem.

There are no known examples of n-types that are not $K(G,1)$s but that are homotopy equivalent to finite dimensional CW-complexes. And that there are no such examples has been stated as a conjecture by Casacuberta-Castellet in "Mislin's book of conjectures":

https://atlas.mat.ub.edu/personals/casac/articles/Mislin.pdf