A generalised probabilistic framework is proposed for reliability assessment and uncertainty quantification under a lack of data. The developed computational tool allows the effect of epistemic uncertainty to be quantified and has been applied to assess the reliability of an electronic circuit and a power transmission network. The strength and weakness of the proposed approach are illustrated by comparison to traditional probabilistic approaches. In the presence of both aleatory and epistemic uncertainty, classic probabilistic approaches may lead to misleading conclusions and a false sense of confidence which may not fully represent the quality of the available information. In contrast, generalised probabilistic approaches are versatile and powerful when linked to a computational tool that permits their applicability to realistic engineering problems.