This punching above their weight is a myth. The only reason we have looked a half-decent Test side in the last 7 years is because we have played very little cricket in tough conditions, and every time we have stepped outside the UAE, we have been exposed.
Lost a Test in WI in 2011, humiliated in SA and even Zimbabwe in 2013, got thrashed by SL in 2014, and then the losing streak in 2016. The only silver lining has been the 2-2 draw in England, but it was a bit of a fluke because we got lucky with the conditions + England had problems of their own.