4. Expand Targeted Runtimes; ensure that "Apache Tomcat v5.5" is present. Click on it. In the Runtime Composition window (second window), you should see:
Apache Tomcat v5.5
JRE 5.0: jdk1.5.0_17 <-- this is where mine, which had been set up for Java 1.4, was wrong
Step four cured my problem. I had imported the project which had been developed on a PC using an older version of Tomcat (that I don't have). Clicking the right version, here, fixed it.