The short answer is situations when a bug can occur only when three or more states must be set a certain way.
The long answer ...
Pairwise testing good technique that actually applies some pretty sound logic. That said, it is fairly easy to understand what the potential tradeoffs are .. let me illustrate using an example I have used previously.
Let’s assume that we are building an application that we use determine the cost of a flight.
We can choose from Sydney, Melbourne or Brisbane as destinations. We can choose from Business, Economy or Budget Economy as our type of ticket. Finally, we can choose Virgin Blue, Qantas or Jetstar as our airline.
Let’s assume that there is a bug which occurs when Business and Jetstar are selected together, because Jetstar don’t have any business class seats. With all 27 possible combinations, we would get three test failures, as shown below:
Destination Class Airline Result
Sydney Business Jetstar Error
Sydney Business Qantas
Sydney Business Virgin Blue
Sydney Economy Jetstar
Sydney Economy Qantas
Sydney Economy Virgin Blue
Sydney Budget Economy Jetstar
Sydney Budget Economy Qantas
Sydney Budget Economy Virgin Blue
Melbourne Business Jetstar Error
Melbourne Business Qantas
Melbourne Business Virgin Blue
Melbourne Economy Jetstar
Melbourne Economy Qantas
Melbourne Economy Virgin Blue
Melbourne Budget Economy Jetstar
Melbourne Budget Economy Qantas
Melbourne Budget Economy Virgin Blue
Brisbane Business Jetstar Error
Brisbane Business Qantas
Brisbane Business Virgin Blue
Brisbane Economy Jetstar
Brisbane Economy Qantas
Brisbane Economy Virgin Blue
Brisbane Budget Economy Jetstar
Brisbane Budget Economy Qantas
Brisbane Budget Economy Virgin Blue
After we apply pairwise (I used allpairs) we reduce to the following 10 tests
Destination Class Airline Result
Sydney Business Jetstar Error
Sydney Economy Qantas
Sydney Budget Economy Virgin Blue
Melbourne Business Virgin Blue
Melbourne Economy Jetstar
Melbourne Budget Economy Qantas
Brisbane Business Qantas
Brisbane Economy Jetstar
Brisbane Economy Virgin Blue
Brisbane Budget Economy Jetstar
So we found our bug in this case, so we didn't loose anything. So what is the trade-off with using pairs?
Well if a bug only occurs when three or more things occur, such as a Qantas business class flight to Melbourne, then you are not guaranteed of finding it. You may get lucky if your test case just happens to match, but you are not guaranteed to find it, that is the tradeoff with pairwise.
You will get this kind of tradeoff with any testing approach, not just pairwise. My example shows what happens with 2 variables, when moving into combinations with more than just pairs, you can increase the coverage.
There are also some tool limits. AllParis (which I used) can only do pairs, whilst other tools will have less limitations.