For a number of years heritage resources have suffered the negative effects of tourist pressure. Overtourism has been generating a wide-ranging debate on this issue. This work focuses on the strategies and measures adopted to address overtourism during the last 30 years in heritage attractions, which are much less studied than urban heritage centres. Specifically, it analyses measures relating to the management of site capacities, based on the concept of tourist carrying capacity, using the content analysis as a methodology. We have used the Alhambra in Granada as a case study, as one of the most visited Spanish World Heritage Sites. The results obtained reflect a strong consensus on the need to introduce measures to restrict tourist activity, although these measures imply a great effort in terms of management and tend to be taken advantage of by some stakeholders for their own benefit. The extensive time frame of the measures adopted allows us to reflect on the possibilities and limitations of dealing with overtourism by using capacity management strategies, and to extract useful conclusions for other heritage sites with similar problems.