Generally, it is not the monitor that would be to blame here.
Most likely, the Video Adapter ("video card") itself is set to display only 16 colors.
To check, go to Start > Settings > Control Panel > Display
Then click the last tab ("Settings").
See what you have in the "Colors" area of that dialog. My guess is that it is set to 16. You can drag it around to adjust the settings. Be careful when changing these settings (make sure the monitor is capable of displaying what you tell it to -- this is more of an issue for resolution that color, however).
Let us know.