There is a setting in ArcGIS server that usually limits returned records to 500 or 1000, you can check this. Also I would make sure the field in question is indexed.
There is a new thread for this subject.
Cheryl,
Peter has some good points there. I sounds like you may have already adjusted the servers max records property if you are getting near 3000 though. This thread was closed in the old forums in favor of the new thread Peter has linked. Some how in the move to GeoNet this thread was reopened.
If you can post your eSearchWidget.xml in this link Enhanced-Search-Widget-for-FlexViewer Part III I can look at the issue.
Timothy Hales is it possible to close a thread to future comments as it was in the old forums? I had gotten Jim to close this thread due to size.
Robert, I have re-closed this discussion.