I've noticed this also. On my home computer I can type my documents in the Run box and it opens the My Documents folder. But on my work computer it doesn't work.
I haven't found what setting turns this off or on, but I played around with it now and noticed the following pattern: You can type the name of any file or folder in your Documents and Settings\username folder without quotation marks, even if it has internal spaces, and it will open correctly.
This suggests the Documents and Settings\username folder may be a default location used if the command entered in the Run box isn't found, and it works even without quotation marks. This clue may help us discover the true setting or reason why this works.
Further tests
I compared the environment variables on my home and work computers and noticed the following difference.
At home, HOMEDRIVE and HOMEPATH together point to C:\Documents and Settings\username
At work, HOMEDRIVE and HOMEPATH together point to X:\ (a networked drive letter). My work computer also has a HOMESHARE variable that indicates the share name of the networked drive letter.
At work, I was able to enter the the names of files or folders in my X:\ folder in the Run box, without quotation marks, even if they had spaces in them, and they opened correctly.
Answer
After some testing, it looks like the Run box searches for matches without requiring quotation marks in the home folder first, then it moves on to searching for matches in the PATH and requiring quotation marks if the name contains spaces.
The correct way to change the home folder is to use the Computer Management tool to modify the properties of the user in question. The environment variables are automatically set by Windows based on the user properties and changing the variables yourself may cause unpredictable results. (That last page is for Windows 2000, but I guess the same advice would apply to Windows XP.)
Workaround
To restore the ability to enter my documents into the Run box, you can put a shortcut in your home folder named My Documents that points to the correct My Documents folder. (To quickly open your home folder, enter . in the Run box.) However, the shortcut doesn't work with additional subfolders on the end of it.