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The answer is no without the monomial ideal hypothesis. I do not know whether it is true with that hypothesis. There exists prime ideals $P\subset S$ such that $S/P$ is not Cohen-Macaulay but, and ideal $J\subset P$ with radical of $J$ equals $P$ and $S/J$ Cohen-Macaulay. Taking $P=I,J$ in your question, gives examples where the above inequality does not hold. It is easy to construct characteristic $p>0$ examples. For a characteristic zero example, see the review of a paper by Cowsik and Nori, MR0393004.