Train Tickets to Shanghai (Beijing)

I am new to Beijing and probably am asking a stupid question to those that have een here for quite a long time, but is it impossible to buy train tickets to Shanghai? I was planning on going to the train station, where ever that is, and hop on a train next Wednesday or Thursday, and people I have spoken to have laughed at me and said that there's no way that there are tickets still available to take the fast train to Shanghai. Does anyone out there know if that's a FACT!? Please share some experience my way. Thanks.