First, if you are traveling on an American passport, the answer is no. If you're a dual citizen (United States and another country), you can probably enter by rail on your second passport, but mentioning that you're an American would be a very bad idea.
There have been rumors of North Korea opening up domestic trains to U.S. citizens, but I haven't seen this confirmed anywhere.
If you're not an American citizen, or at least not traveling on an U.S. passport, the answer is yes. There are a few tour operators who organize train trips to the DPRK, but it should also be possible to buy a train ticket in Beijing directly. You can find information on the train times here:
http://www.seat61.com/NorthKorea.htm#.Ux-mMV46ntk
It appears that the train runs four times a week and is usually reliable.
Be sure to obtain a visa for travel to North Korea before departing.
As a previous answer mentioned, the story of two Austrians who used the Moscow-Pyongyang sleeper car is widely available on the web, including at http://vienna-pyongyang.blogspot.com/
I thought their journey was fascinating (and extremely dangerous), but I wouldn't recommend attempting to use this route today.
Edit: Air Koryo does indeed have a terrible reputation. However, they've purchased a few new planes which are running on the Beijing-Pyongyang route. Their Tu-204 aircraft are now approved to fly into the European Union, though they don't operate a route to Europe at this time.
A better option than Air Koryo is Air China, which does operate to Pyongyang several times per week. They fly a modern fleet which is considered quite safe.