Basically the same thing Bill said above about the rubber- my HCG vials have a lot of thick rubber on the inside, to where if you don't get it just right in the center, it won't draw. However, the fact that you consistently get this problem after 2-3 pins (I assume multiple different vials?) then it seems something else odd is up. Maybe try a different needle.
Try to draw it out with a larger gauge, 1" needle, if it works just load that syringe up to max (3ml would last several shots) and then backfill into your smaller syringe. Pull some extra air into the "storage syringe" and try to store with the needle pointing up in your fridge to prevent any leakage...few drops at most.