While every customer deserves to be responded to individually, in a personalized manner, those individual, personalized responses are going to suffer across the board if you are making some underlying, structural mistakes as an organization, including the following frequent offenders:
Failures in talent management and HR
Nothing detracts more from the customer experience than an ongoing failure to recruit effectively, hire carefully, and manage employees for professional and personal growth. In very broad strokes, you should be striving to hire for personality traits that are conducive to customer-facing work, and to do so as scientifically as possible, ideally making use of one of the scientifically verified screening tools that are available. (These tools, however, generally should not be trusted until you have verified them yourself as matching your own organization's goals and its experiences to date.)
Control and empowerment issues
Almost as bad as failure to hire the right people is failure to give them enough control over the design of their jobs, and empowerment in their interactions with customers. If you exert excessive control in these two areas, you reduce what you get from your employees; how can they contribute their best to an organization that keeps asking them to do otherwise?
Employees who are untrained or undertrained in customer service are accidents waiting to happen—and you probably won't have to wait long. Watch out as well for the danger of mis-training employees; a rigidly old-school customer service trainer who schools them to be more formalistic and scripted than customers expect can be almost as much of a problem. Customers today want great customer service, but they also want genuine service style.
Failure to plan for customer service recovery
Customer service recovery is an essential part of great customer service. It's essential to have a plan for service recovery in place, trained on, and socialized throughout your company before the customer hits the roof and succeeds in getting you flustered. If you aren't currently deploying your own system, I offer my own AWARE system at the following link.
Using yesterday's standards for speed
Incredibly, I still run into companies which have 24-hour standards (and more that have a 12-hour standard) for returning customer emails and phone calls. For a customer today, waiting 24 (or even 12) hours for a response feels like years in internet time! After that amount of time, a customer's likely to assume you're never going to get back to them–and they'll find another company who will.
Lack of a continuous improvement and innovation mindset
Even the best-intentioned companies are going to get stuck in their ways if they don't systematically encourage innovation. It's not enough to be good enough today; "good enough" will quickly become "less than" if you don't involve your entire organization in innovation–both of the bold-leap variety and in terms of continuous improvement.