Familiarity breeds contempt, and the extensive dissemination of information on cointegration tests is a case in point: their ready availability in both econometric and trading software has led to their frequent misapplication. It's easy to assume that since pairs trading has grown rapidly in popularity in the retail space that this is where most of this misapplication takes place. Some of it almost certainly does, but many professional trading operations have also tended to make easy assumptions about the tests, without really taking the trouble to understand what is going on under the hood. This understanding doesn't need to be at a microscopic level, but should at least extend to knowing exactly which flavour of the test is appropriate for a particular purpose and understanding what any results really mean in practical terms.
Built for something else
The first thing to say is that the cointegration tests commonly used in trading today were never originally intended for that purpose. Instead, they have their origins in the world of economics where the concept of cointegration was created to model long run relationships between economic factors. In many cases there was a credible theoretical underpinning for these relationships, which is very seldom the case in the way they are applied in most trading scenarios.