The relationship between efficiency, quality, and access in healthcare is far from being well defined. In particular, there is no consensus on whether there is a trade-off between hospital performance and its social dimensions, such as the care appropriateness, safety, and access to proper health care. This study proposes a new approach based on the Network Data Envelopment Analysis (NDEA) to evaluate the existence of potential trade-offs between efficiency, quality, and access. The aim is to contribute for the heated debate around this topic with a novel approach. The suggested methodology combines a NDEA model with the weak disposability of outputs to handle with undesirable outputs related to the poor quality of care or the lack of access to appropriate and safe care. This combination results in a more realistic approach that has not yet been used to investigate this topic. We utilised data of the Portuguese National Health Service from 2016 to 2019, with four models and nineteen variables selected to quantify the efficiency, quality, and access to public hospital care in Portugal. A baseline efficiency score was calculated and compared with the performance scores obtained under two hypothetical scenarios to quantify the impact of each quality/access-related dimension on efficiency. The first scenario considers that each variable, individually, is at its best situation (for example, absence of septicaemia cases), and the second one, at its worst (e.g., all seen inpatients had a septicaemia case). The findings suggest that there might exist meaningful trade-offs between efficiency, quality, and access. Most variables exhibited a considerable and negative impact on the overall hospital efficiency. That is, we may expect a trade-off between efficiency and quality/access.