CEDET

One clue seems to be this.  In semantic-analyze-possible-completions and
semantic-analyze-possible-completions-default in complete.el, the argument
to the `context' parameter seems to be missing a value in the :prefixtypes
slot, when completing on the variable `s' in Client.java.  It then bails
out of semantic-analyze-possible-completions-default beginning at line
144.

Question:  where does the argument to `context' come from when invoking
semantic-analyze-possible-completions, say, interactively?

Yet another clue.  If instead of trying to complete on variable `s'--whose
declared type is Server--and instead try to complete on an object literal
for that type, Analyzer completion works.  That is, if instead of this:

Server s = new Server();
s.  <-- try to complete here

I do this:

(new Server()).  <-- try to complete here

This seems to work.  Interestingly, if I try to complete on the type, but
not on a new object literal of that type, instead of getting just the class
variables (static variables and methods in Java-speak), completion "works",
but it gives the wrong results.  For example:

Server.  <-- try to complete here

This SHOULD only complete class (or "static") variables and methods.  It
should filter out instance variables and methods, but it does not.  This
may be a separate problem.

Presumably, these things work perfectly in C++ (though I haven't checked),
since CEDET and Semantic seem to have the most robust support for C/C++.
In the syntax and semantics, Java is a pretty close match to C++, so I'm
sort of surprised this doesn't work.