If the question specifically asks if you busted an FAA check ride, it looks like you can say NO. If this is an application for an airline job, you need to disclose failing any job-related training, and they will do a background check to verify what you say.
The purpose of the check ride background check is to protect passengers from a bad pilot. Years ago, pilots at the controls of several regional airline accidents had come to the regional airlines after getting FIRED for busting LOTS of check rides at other airlines. Now airlines do extensive background checks on your flight training history so they can prevent this from happening again.
The chief pilot where you used to work will get a 10 page form to fill out that specifically asks if you ever failed any training at that place. Looks like you already know the answer. Perhaps you could call him to find out what he will say so you can understand how negative it will be. Who knows, maybe he liked to bust everyone when they came in for initial training, sort of a technique to get them off their high horses.
Last year, I remember reading an article in AOPA Flight Training magazine about what NOT to do in a regional airline interview. It specifically addressed this issue and gave some tips on how to handle busted check rides so you might want to read it.
