What I've found on my system (RH Enterprise 3, CrossOver Office 3.0) is that the installation seems to work fine, but any applet you try to run won't pop up fully, and the IE browser freezes hard.
Right before the crash, in the upper left of the screen where the applet seems like it's being initialized, a small window pops up; the window is tiny and seems to have no data in it.
It doesn't seem to matter whether you install using the standard web interface or by downloading the j2re-1_4_2_04-windows-i586-p.exe standalone executable; both do the same thing, except that the web installer automatically redirects you to a test page at the end with an applet which kills you right on the spot, while the executable just exits normally.
I have just posted a long trace of what's going wrong in the "Known Issues" area of this section, see there for details. It appears this problem is due to DirectDraw calls not implemented in Wine yet that the Java for Windows code uses.