What happens when high blood pressure won't go down even when taking medication?

Depends. What med(s) are you on? Are you at maximum dose of each? How high is the blood pressure? Are you overweight? Are you diabetic? Docs have a number of different BP meds that work by different mechanisms; so they can change meds, increase doses, add another med. Exercise & eat healthful meals to attain proper weight & relieve stress.
BP hi on meds. You have what is called drug resistant HTN. Often do to a specific cause. Most can be easily tested for. That is what I do and happy to help. .