CRP is released as an acute phase reactant in response to IL-6, where “acute” is a key word: if the problem is new and active (eg moderately severe cellulitis, myocardial infarction), cell biology tells us that the CRP should be elevated. Most of the infectious disease specialists I’ve met check CRP at least once to help guide management of bacterial infections. In my own practice I routinely obtain serial CRPs if the WBC is not dropping as quickly as I want it to (there are other non-inflammatory causes of WBC elevation, but not so for CRP). Interesting tidbit as a sign of the times: in patients with COVID-19 who are treated with monoclonal antibodies like tocilizumab (which is an IL-6 antagonist), a dramatic reduction in CRP tends to herald a clinical improvement.
Now consider the erythrocyte sedimentation rate (ESR). ESR is related to how many circulating proteins are bound to the erythrocyte cell membrane, so rather than measuring acute inflammation I consider it more of an integral of inflammation over the lifespan of the cell, about 4 months. For patients with chronic or subacute inflammation (eg osteomyelitis, rheumatoid arthritis), the ESR is often elevated with a normal CRP. In those conditions it is still true that we want it to normalize with therapy, and if it remains elevated it requires an explanation.
I am not aware of any systemic inflammatory conditions that are routinely associated with a normal CRP and ESR. These two tests together form a compelling argument that acute and chronic inflammation is not present. To be fair, I’m not sure how one would even define inflammation in that case.