I'm kind of surprised that we don't have a resident "bird guru". Most of us have random bird photos in our portfolios, but I haven't seen anyone who is an outright bird fanatic.
Here is a pretty comprehensive list of bird photography tips. That said, there's some sound advice here already. 1/160 is too slow for hand held photography in general, and especially for birds. 1/500 should be a minimum, and with a 300mm lens that's really going to be pushing it, especially when your subjects are small like birds.
Nikon D5000
Shutter: 1/160
Aperture: f/16
ISO 800
With EXIF data like that, I'm inclined to believe you're either shooting in Auto or AP mode. f/16 is way too high for bird photography, as you typically want to control the DoF as much as possible. The exposure triangle is a juggling act, but mastering it will make you a better photographer. Next time, try a couple things:
AP Mode - Twilight
Switch your mode dial to "A" and set your aperture as low as you can. Then meter your scene and check other settings. Is your shutter speed 1/500 or faster? Are you at ISO 200?
SP Mode - Daylight
Switch your mode dial to "S" and set your shutter speed to 1/500. Again, meter your scene and check the aperture and ISO.
Personally, I would start with shutter speed (especially if you don't own a tripod). During normal daylight hours, you should be able to crank up shutter speed into the 1/1000 or above range and still have a shallow enough DoF.