You can draw blood from the "IV arm" below the intravenous needle if you put a tourniquet in between.

The authors performed a comparative study of the values of 18 serum constituents of blood specimens taken simultaneously from the arm into which an intravenous solution was flowing, using a site distal to the intravenous needle, and from the other arm. Subjects of the study were 15 patients at the Norwalk Hospital. Statistical analysis showed no clinically significant difference between the levels in the two arms except for elevation of glucose in the "iv arm" when the intravenous solution contained glucose.