[This answers Reid's petition for an example in the comments, in answer form so as to be able to preview]
Stallings has given in [Stallings, John. A finitely presented group whose 3-dimensional integral homology is not finitely generated. Amer. J. Math. 85 1963 541--543. MR0158917] an example of a finitely presented group $G$ such that $H_3(G)$ is not finitely generated. It follows from this that the 3-skeleton of $BG$ is infinite. The group is $$G=\langle a,b,c,x,y:[x,a], [y,a],[x,b],[y,b],[a^{-1}x,c],[a^{-1}y,c],[b^{-1}a,c]\rangle.$$ Stallings' paper is characteristically short and beautiful, and the proof is a nice application of Mayer-Vietoris (!)