Well I've tried running both the web-based and the downloadable client numerous times, including reinstalls and the same thing always happens. The launcher/game opens with a black screen and closes after a couple of seconds.
I am running Windows XP SP3 32-bit, I have the latest version of Java and the latest graphics drivers. My CPU is a Pentium Dual Core E5400 (2.7GHz), I have 3.25GB RAM and my graphics card is an ATI Radeon X1900, both well within the recommended requirements for the game. I have even opened up the listed ports on my router and software firewalls and the game still closes.
Here is the error log: http://pastebin.com/Y8xTPt1q
My friend also has an X1950 (same line of card) and his computer does the same exact thing. Windows 7 x64, Latest Java. We've gone through the troubleshooting suggestions here, even the shotgun method. Changed out the Video card for an Nvidia 7900GS, and now it runs perfectly. Has something to do with the ATI X1900 range of cards, it seems to freak out Aero pretty badly too when it crashes.