Assume that $\omega<\kappa_1<\dots< \kappa_n$ are infinite cardinals such that for each $1\le i\le n$ there is a $\kappa_i$-complete, $\kappa_i^+$-saturated ideal $\mathcal I_i\subset \mathcal P(\kappa_i)$. Can you obtain a ZFC models which contains $n$-many measurable cardinals? The natural candidate is $L[\mathcal I_1,\dots, \mathcal I_n]$.

It is well known that the answer is yes for $n=1$ (see Kunen: Some applications of iterated ultrapowers in set theory. Ann. Math. Logic 1 (1970), 179–227.)