Of course I find the answer right after asking the question, even though I searched for quite awhile before...
The problem is that Cobertura has trouble working with Java 1.7. You must add the following line to your pom.xml:
<argLine>-XX:-UseSplitVerifier</argLine>
That goes in the configuration element. Here is the entire Cobertura section:
<plugin>
<groupId>org.codehaus.mojo</groupId>
<artifactId>cobertura-maven-plugin</artifactId>
<version>2.5.1</version>
<configuration>
<argLine>-XX:-UseSplitVerifier</argLine>
<formats>
<format>xml</format>
</formats>
</configuration>
<executions>
<execution>
<phase>test</phase>
<goals>
<goal>cobertura</goal>
</goals>
</execution>
</executions>
</plugin>
Now everything works as expected.