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:
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":