Transport for London’s pay as you go transport scheme promises “Our aim is to ensure that Oyster always charges the lowest fare. Where it doesn’t we will refund the difference.” 
The myriad of different prices  and types of journey (tube, national rail, through journeys – ) put most people off from actually working out how much their journey would cost with each leg paid separately. The combination of complex rules and pricing, along with a price promise hide the fact that in many cases it would work out cheaper to use two Oyster cards instead of one.
Three common situations catch people out:
1. To peak or not to peak: If you travel part peak, part non peak, part National rail, part tube, it will be cheaper to have two Oyster cards and use one for the peak, one for the off peak part of your journey.
2. Two zones, one station: If you travel via an interchange which is in two zones, such as Vauxhall (in zone 1 and 2), it will again be cheaper to use two Oyster cards – one for a journey within zone 1, another not touching zone 1.
3. One peak, and your cap shoots up: If you travel on only one peak journey in a day, that will bump your daily price cap up to the peak travelcard, which can be double the price of an off peak one. By doing a £1.80 peak single in zone 1 on one Oyster in the morning peak, then unlimited off-peak travel between zones 1 to 6 on a separate Oyster saves £5.50 a day, compared to just using one Oyster card.
TfL do provide an alternative fare calculator tucked away within the single fare calculator:  but it’s easily missed, and ignores the impact of doing part of the journey peak and part of the journey off peak.
In a beautiful moment of irony, calling the 0845 number given in the leaflet to enquire about the price promise, the call centre agent advised against having two Oyster cards as “you might get confused” (examples provided by @yousability).