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A conceptual useful way to think about this issue is to consider $J=I^{(n)}$, the $n$-symbolic power of $I$, which by definition is the $I$-primary component of $I^n$.

When $R$ is a polynomial rings over $\mathbb C$, this is the ideal consisting of functions vanishing to order at least $n$ on $X = \text{Spec}(R/I)$.

It is then well-known that the depth of $R/I^{(n)}$ can go down. For example, take $I$ generated by the $2\times2$ minors of the generic $2\times 3$ matrix inside the polynomial rings of the $6$ variables, localized at the maximal ideal of those variables. Then $R/I$ is Cohen-Macaulay of dimension $4$, but $R/I^{(n)}$ would have depth $3$ eventually. For more general statements about ideals of maximal minors, see for example Section 3 of:

Powers of Ideals Generated by Weak d-Sequences, C. Huneke, J, Algebra, 68 (1981), 471-509.

EDIT: the example above looks specific, but such examples should be abound. I expect most Cohen-Macaulay ideals which are not complete intersections to give an example (it is known that $R/I^n$, the ordinary powers, are CM for all $n>0$ iff $I$ is a complete intersection). The $2\times 2$ minors gives is a generic situation of non-complete intersection but CM ideal.

A philosophical comment: it is unlikely that Cohen-Macaulayness will be preserved by basic operations on ideals. So if $R/I, R/J$ are CM, we do not expect $R/\sqrt{I}, R/I^n, R/I^{(n)}, R/P$ ($P$ an associated primes), or $R/(I+J), R/IJ$ etc. to be CM.

The reason is that to preserve depth one needs to control the associated primes, and these operations only allow you to control the support. However, finding an explicit example is usually not so obvious.

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A conceptual way to think about this issue is to consider $J=I^{(n)}$, the $n$-symbolic power of $I$, which by definition is the $I$-primary component of $I^n$.

When $R$ is a polynomial rings over $\mathbb C$, this is the ideal consisting of functions vanishing to order at least $n$ on $X = \text{Spec}(R/I)$.

It is then well-known that the depth of $R/I^{(n)}$ can go down. For example, take $I$ generated by the $2\times2$ minors of the generic $2\times 3$ matrix inside the polynomial rings of the $6$ variables, localized at the maximal ideal. Then $R/I$ is Cohen-Macaulay of dimension $4$, but $R/I^{(n)}$ would have depth $3$ eventually. For more general statements about ideals of maximal minors, see for example Section 3 of:

Powers of Ideals Generated by Weak d-Sequences, C. Huneke, J, Algebra, 68 (1981), 471-509.