I configured fitnesse-jenkins plugin to run a suite from already running Fitnesse server. And seems Jenkins runs the test, saves the result. However it reports the build as SUCCES despite of the fact that most tests have failed - i.e. the output contains:
<finalCounts>
<right>54</right>
<wrong>175</wrong>
<ignores>1</ignores>
<exceptions>18</exceptions>
</finalCounts>

Here is the console output of the Jenkins build job:

Started by user anonymous
Building in workspace /home/me/.jenkins/jobs/MyDemoJob1/workspace
hudson.plugins.fitnesse.FitnesseBuilder:

{fitnesseTargetPage=FitNesse.SuiteAcceptanceTests, fitnesseTargetIsSuite=true, fitnesseHost=localhost, fitnesseHttpTimeout=180000, fitnessePortRemote=8980, fitnesseStart=False, fitnessePathToXmlResultsOut=fitnesse-results.xml}

Connnecting to http://localhost:8980/FitNesse.SuiteAcceptanceTests?suite&format=xml&includehtml
Connected: 200/OK
4k...
...
4264k...
Xml results saved as UTF-8 to /home/me/.jenkins/jobs/MyDemoJob1/workspace/fitnesse-results.xml
Finished: SUCCESS

I'm attaching the generated fitnesse-results.xml