I discussed this with Martin G. for a while and he
asked that I file it.  Here is all the info we
were able to collect in the email thread.

Tomcat is started from the IDE by the test. 
However, I believe it is also stopped.  The end of
the case does: Debug->Finish and hits okay on the
Finish Debugging Sessions dialog
It then tries to Stop Tomcat if necessary by going
to the runtime tab, navigating to Server
Registry->Tomcat...->http and checking whether
it's running.  I think it decides in the actual
run that this is not necessary after the finish
debug, but let me know if it should be.

From Martin - Finish debugger does not stop Tomcat
- it only detaches debugger from it. If Tomcat
wasn't stopped before, then it should really be
stopped when IDE exits - so there may be some
problem there...does it disappear after a minute?
What happens if you try to open an url on that
Tomcat (like, http://localhost:8084/ if 8084 is
the port on which it is running) - does it open a
page or you get some error page - what error?

From me: I launched a browser and put in
http://localhost:8084/MyJSP.jsp which is my jsp
and it brought back the data - so that is proof
that it's still running.  The process doesn't
disappear unless I kill it.  Here's the ps info: 
/opt/jdk/1.4.2/SunOS//bin/java -Xdebug -Xrunjdwp
-Djava.

It is truncated, but shows the debug still there.