Can I use a 50 watt bulb in a 20 watt desk lamp?

Most like result would be melting the socket and it's electrical parts. Might work, but it isn't really worth it.

Who the heck designed a 20W lamp, anyway?

Answer: An IKEA Smurf who wanted to get you back into the grotto and sell you something else.

A 3W LED-bulb replaces a 20W halogen-bulb.

These are high intensity halogen lamps. They generate a lot of heat...far different than a regular incandescent bulb. Don't change from a 20 wt halogen bulb as there's a good chance doing so could create overheating and/or melting or catching fire of the insulation on the wiring and/or melting of the socket. This warning is not just to warn you for legal reasons.
http://lighting.about.com/od/Fixtures/fl/Donrsquot-Overlamp-Your-Fixture.htm

There are two risks.

First, the bulb will generate more heat than was intended for the enclosure, which will make the enclosure much hotter than intended by design.

If the enclosure is metal, you may be able to mitigate this risk by not touching the enclosure or letting flammable materials come into contact with it. If the enclosure is plastic, there is a possibility of it melting or catching on fire.


Second, the power cord will likely have a gauge of wire that is only designed for a 20 watt bulb. Running enough current for a 50 watt bulb will cause your power cord to get abnormally hot.

If it gets hot enough, it could cause a fire directly. If not hot enough to cause a fire directly, the heat could still cause the insulation to get brittle and break, leading to a short circuit that could cause a fire.