There's an obstruction to embedding $n$-complexes in $\mathbb R^{2n}$ provided $n \geq 3$ due to Shapiro. 

MR0089410 (19,671a)
Shapiro, Arnold
Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction.
Ann. of Math. (2) 66 (1957), 256–269. 

or as Ian suggests, van Kampen pre-dates Shapiro. 

JFM 58.0615.03 van Kampen, E. R.
Berichtung zu:``Komplexe in euklidischen Räumen''. (German)
Abhandlungen Hamburg 9, 152-153 (1932).

Could you be more precise on how you want to modify the your complexes that originally are in $\mathbb R^3$, your special case?