For one of the links that the Broken Link Checker is reporting as broken even though your client knows is not broken, can you go into the WordPress Dashboard under "Tools->Broken Links..." and expand the error message log for that link and copy the full error log details and paste them back here?
This log message details will point us in the right direction where to look next to see what is going wrong. For example if in the log one sees this line:
" Log : Could not resolve host: suchandsuch.com [Error #6]"
the text and the "Error #__" number will tell us which curl error message is being generated which would tell us where to look next.
( by referencing this curl error list )
(sample output here -- please post corresponding/similar info. copied from one of the false positive reported broken links on your client's site)
URL Status Link Text Source
Server Not Found
bonfjfjkdjdfkj.com hello testing the link to bonfjfjkdjdfkj.com to see if it is broken
Link last checked : July 30, 2013
HTTP code : 0
Response time : 0.003 seconds
Final URL : http://bonfjfjkdjdfkj.com/
Redirect count : 0
Instance count : 1
This link has failed 2 times.
This link has been broken for 45 minutes.
Log : Could not resolve host: bonfjfjkdjdfkj.com [Error #6]
=== (No response) ===
Link is broken.