The two elements can go at the same time. If the continuity tester says that they are both OC then it would not surprise me, my wife is an expert in getting them to do this. Often you do not notice that the first one has popped anyway.
Many of the cheaper tumble dryers are made by Crosslee in Halifax, if you look at the back of many dryers you will see that they all look the same.
See http://www.crosslee.co.uk/electric.html
Peter C
June 2004