Nice post from CPA Success illustrates the point: It's not just when things go "right" with clients but what happens when things go wrong? .... That is when you can build up two big things: TRUST and LOYALTY.
Trust because if a client KNOWs you take "mess-ups" very seriously, they know they will be taken care of to the best of your ability. They also know you care a lot when it happens. Loyalty is built when your "mess-up" becomes memorable .... IN THE WAY YOU RALLIED TO THEIR SIDE TO TAKE OWNERSHIP. Everyone is human. We all know we are not perfect, but we all expect certain fundamnetal things to happen when a vendor "messes up", but it rarely happens.
And the first place to start, and sometimes the hardest ....
Admit the mistake. Admit it. Come clean. Be blunt. Be serious. People "rarely" do this, but we all crave it.
By doing that, you will show them you take them (and your service to them) very seriously!