Terje, I shot quite a few frames of black capped chickadees here in Fairbanks recently with my 300PF on D7100. These are probably just as fast as yours, and were filling about the same part of the frame. Usually VR was on at 1/640 - 1/1000 second (although it is recommended to turn it off in that shutter speed range), ISO 1600-3200. The "keeper" rate was surprisingly high as long as I managed to get the focus point near the eye in that fraction of a second that was available. Most non-keepers had the bird out of the frame when it took off, or the bird had its back turned toward me so there wasn't enough to focus on to get the eye sharp. (Some also had too slow shutter speed.) Light levels were pretty poor, often at dusk. No AF tuning applied.
Is it possible that the 200-500 just does not focus fast enough for these small birds? Did you have AF-C configured with shutter priority (and AF-on) which is pretty standard for these kind of applications. Then the shutter will of course fire even if the lens has not finished focusing. However this should be possible to detect as it would lead to either back focus or front focus depending of the direction the bird moves.
