Nikon has put up a service advisory regarding the D5000. It has to do with a power problem that affects a limited production series of this camera, which may not properly power on, even with a fully charged EN-EL9a battery or connected to AC with the EH-5a power adapter.
Affected cameras can be identified through their serial number (located at the bottom of the body - see picture below) and Nikon has set up a special service webpage to allow D5000 owners to determine if their camera is part of this particular batch.
In the case of owing an affected model, owners should contact their local Nikon customer support to obtain a free repair.
Click here for more information, and here to get to the page for verifying your serial number.
Nikon have added new serial numbers to this list as of August 12, and recommend users who checked their serial number previous to this date, to check again.