I think the trouble is that the keyboard's buffer can't handle that many keys and so it won't actually tell the operating system when a new key is pressed. I don't think there's any way around it. I have noticed that some keyboards are worse than others however and some keys seem to fill the buffer quicker than others. I'm pretty sure this isn't a java-only thing but happens in all games & programs.
Last time I wrote a two-player game (ten years ago!) I ran up against this problem. My keyboard seemed to be the only one on the planet on which the game was playable. (That was with up, down, left, right and fire keys for each player, and players would often be holding down three keys at once -- two directions plus fire.)
As far as I can tell, the severity of the problem varies from keyboard to keyboard (the keyboard on my current laptop is particularly bad for it). Some key combinations are worse than others, but I've noticed that there's less of a problem for the 'special' keys such as Shift, Alt, Ctrl and the Cursor keys. (If I remember correctly Doom used to use those keys -- maybe not a coincidence?) That said, you may run into other problems if you use those keys in a game because some operating systems (or users) assign special extra functionality to them (for example, if you hold down the right shift key for long enough on Windows, unexpected things happen).
The solution? Provide alternative playing keys, or user-definable keys. And avoid writing two-player games!
BUT... The original example was talking about a two-player Pong game, which presumably wouldn't expect the players to be pressing more than one key each at a time. So in that case I'm guessing that the problem is more likely to be the way that the code is processing the key events (as woogley said), rather than the events that the keyboard is sending.