A quick Google throws up http://ubuntuforums.org/showthread.php?t=54181
(amongst others) where it is suggested that you either simlply delete
~/.ICEauthority or boot to a terminal and check that the owner/group are
set to you and that you have rw permissions.

There doesn't seem to be any consensus as to the cause though, which is
annoying.

I did a ls -l on that ICEauthority and it apparently belonged to root. I did
the sudo chown to change it to me. I have yet to reboot to see if that fixed
it, though I suspect it did. Was changing the ownership to me the right thing
to do? I'm the only one who uses my computer.

Yes, that file should be owned by the user who's home directory it is in
- in your case, you - not root. I can't find any information on how it
becomes owned by root, but it seems to be a reasonably common problem.

You should also check the group it's in, which should also be you,
although any change here should have no impact on operation as the
permissions should be -rw------- (ie you have read/write permissions
while the group and global permissions are no access at all).

That did, indeed, fix the problem. I can boot into other DEs now. I don't
remember what message popped up just before the problem started. I was booting
and some error message popped up, and I clicked OK to continue because I had
something to do on the puter, and I figured if it causes a problem, I'll unwind
it later. The boot after that one was when the ICEauthority error started
happening.

I'm trying to get ahead of the curve here with the new DE replacing GNOME with
the next version. I know I've been told which it is, but I don't remember. Which
DE is replacing GNOME?