In Tent, Ziegler: A Course in Model Theory it is stated on page 89, that
Saturated structures need not exist (think about why not), but by considering special models instead, we can preserve many of the important properties – and prove their existence.
I do not see why this statement holds in this generality. Finite structures are always saturated, so consider a (complete) $L$-theory $T$ with infinite models. By Löwenheim-Skolem we find a model $M\vDash T$ with $|M|=\kappa\geq |T|^+$. Then for $A\subseteq M$ with $|A|\leq|T|$ we have $|S_n^M(A)|\leq 2^{|T|}$. If we assume the continuum hypothesis $2^{|T|}=|T|^+$, then we may add at most $|T|^+=2^{|T|}$ additional elements to realize all omitted types of $S_n^M(A)$ to the structure $M$ and by compactness get an elementary extension $N\succ M$ of the same cardinality. So it should always be possible to find a saturated model at some sufficiently large cardinality in this setting.
Is my guess that it is about the continuum hypothesis correct or am I missing the point?