Can you change what a symlink points to after it is created?

Yes, you can!

$ ln -sfn source_file_or_directory_name softlink_name

AFAIK, no, you can't. You have to remove it and recreate it. Actually, you can overwrite a symlink and thus update the pathname referenced by it:

$ ln -s .bashrc test
$ ls -al test
lrwxrwxrwx 1 pascal pascal 7 2009-09-23 17:12 test -> .bashrc
$ ln -s .profile test
ln: creating symbolic link `test': File exists
$ ln -s -f .profile test
$ ls -al test
lrwxrwxrwx 1 pascal pascal 8 2009-09-23 17:12 test -> .profile

EDIT: As the OP pointed out in a comment, using the --force option will make ln perform a system call to unlink() before symlink(). Below, the output of strace on my linux box proving it:

$ strace -o /tmp/output.txt ln -s -f .bash_aliases test
$ grep -C3 ^unlink /tmp/output.txt 
lstat64("test", {st_mode=S_IFLNK|0777, st_size=7, ...}) = 0
stat64(".bash_aliases", {st_mode=S_IFREG|0644, st_size=2043, ...}) = 0
symlink(".bash_aliases", "test")        = -1 EEXIST (File exists)
unlink("test")                          = 0
symlink(".bash_aliases", "test")        = 0
close(0)                                = 0
close(1)                                = 0

So I guess the final answer is "no".

EDIT: The following is copied from Arto Bendiken's answer over on unix.stackexchange.com, circa 2016.

This can indeed be done atomically with rename(2), by first creating the new symlink under a temporary name and then cleanly overwriting the old symlink in one go. As the man page states:

If newpath refers to a symbolic link the link will be overwritten.

In the shell, you would do this with mv -T as follows:

$ mkdir a b
$ ln -s a z
$ ln -s b z.new
$ mv -T z.new z

You can strace that last command to make sure it is indeed using rename(2) under the hood:

$ strace mv -T z.new z
lstat64("z.new", {st_mode=S_IFLNK|0777, st_size=1, ...}) = 0
lstat64("z", {st_mode=S_IFLNK|0777, st_size=1, ...}) = 0
rename("z.new", "z")                    = 0

Note that in the above, both mv -T and strace are Linux-specific.

On FreeBSD, use mv -h alternately.

Editor's note: This is how Capistrano has done it for years now, ever since ~2.15. See this pull request.