I did not have a servlet-api.jar in the application's WEB-INF/lib directory (or any other jar there). The problem was that in WEB-INF/web.xml I had this...
...
<servlet>
<servlet-name>MyServlet</servlet-name>
<servlet-class>com.bizooka.net.MyServlet</servlet-class>
</servlet>
<servlet-mapping>
<servlet-name>MyServlet</servlet-name>
<url-pattern>/MyServlet</url-pattern>
</servlet-mapping>
...
...according to a tutorial. Removing that still allowed the servlet to function, and did not get the error.