I frequently use this saying when watching public figures get in trouble. And, given the number of scandals that have been elevated to '-gate' status because of it, you'd think that people would learn by now to just "fess up". Unfortunately in our jobs this is just as true in the public arena. How many times have you 'swept a problem under the rug', spent as much time covering up an issue as it would to just admit it and fix it.
If you are an employee -- admit it, fix-it, make changes to prevent it, and move on. If you keep messing it up, no matter how well you cover it up, it will get out anyway.
If you are an employer --create an environment when your employees can do as I suggested.
Both employees and employers -- This does not mean that if there are perpetual issues they shouldn't be dealt with, everyone needs improve. Organizations don't stay in business if everyone isn't doing the their job.