Assume one is given a commutative square of spaces
$A \quad \to \quad C$
$ \downarrow \qquad \qquad \downarrow$
$B\quad \to \quad X$
which is a pushout and in which each map is a cofibration. If $A \to B$ is $r$-connected and $A\to C$ is $s$-connected, then the Blakers-Massey theorem says that the square is $(r+s-1)$-cartesian (this means that the map from $A$ into the homotopy pullback of the remaining terms is $(r+s-1)$-connected).
The only proofs of the statement that I know of (at this level of generality) make use of transversality. However, if all spaces are simply connected, there are proofs which avoid transversality (for example, when $B$ is a contractible, one can deduce it using the Serre exact sequence).
Question: Is transversality intrinsic to a proof of the theorem in the general case?