Could you request to be put to sleep during a bone marrow biopsy?

Anesthesia. Yes, you could have an anesthesiologist involved if you arranged this with the physician ordering the test.
SURE. A local anesthetic is often enough to make the discomfort tolerable. However, there are people who fear closed spaces so take a tranquilizer and doze off during the procedure. The same with people who are afraid of needles. If the test is necessary, the doctor should make sure that you are able to tolerate it. If they aren't sympathetic, get another doctor.