Think about what happens to a piece of glass when you hit it too hard: it shatters into a million pieces. We say that glass is brittle because it breaks before it bends.
Not all materials do this. Steel, for example, is likely to bend (maybe a lot) before it actually breaks. This is why we build bridges out of steel and not glass.
It's useful to apply the concept of brittleness to the world of business. Customer experiences, like bridges, are designed to handle a certain amount of strain before they start to fail.
When things start to go wrong, a brittle experience is likely to go catastrophically wrong for the customer or the company (or both). On the other hand, if the process is flexible enough to bend a bit and handle the unusual situation, it may not be that big of a deal.
For example, air travel today is very often a brittle experience. When everything goes well (as it usually does), you get to your destination on-time and with at least some dignity intact.
But if your travel plans go even slightly awry, the airline experience quickly goes from smooth to a stressful mess which could extend longer than the original trip. A brief thunderstorm at your departure airport means there's a long line of planes waiting to take off, and you sit on the ground for an hour or two. That departure delay means you miss your connecting flight. The next flight to your destination is overbooked, so you wind up spending the night at your connecting city waiting for a flight with an open seat to take you to your destination.
What started out as a minor hiccup (the brief thunderstorm) quickly turned into a stressful multi-day experience because the airports and airlines are too overloaded and too inflexible to handle even minor disruptions without it spiraling out of control. That's brittle: small problems become big problems and the whole thing goes very wrong for some passengers.
It's worth examining all elements of the customer experience under the lens of brittleness. Of course we expect that most of the time things will go smoothly for most customers, so the "normal" experience needs the most attention. But even the best-designed system won't be able to handle every situation.
So what happens when a customer has a problem? Are you flexible enough to deal with it gracefully? Or does the customer experience shatter into a million pieces like a piece of glass?