The importance of stimulating greater sharing of data for use and reuse in health research is widely recognized. To this end, the findable, accessible, interoperable, and reusable (FAIR) principles for data have been developed and widely accepted in the research community. Research biospecimens are a resource that leads to much of this health research data but are also a form of data. Therefore, the FAIR principles should apply to biospecimens. Nevertheless, there is a widespread problem of not sharing biospecimen resources that is clearly visible within the research arena. The impacts of this are likely to include diversion of precious research funds into compiling duplicate biospecimen cohorts, detraction from research productivity as researchers compete for and create duplicate resources, and deterrence of attempts to assess research reproducibility. This article explores some of the barriers that may limit availability of FAIR biospecimens. These barriers relate to the type of biospecimen collections and the characteristics of the custodians that influence their intention and interest in sharing. Barriers also relate to the ethical, legal, and social issues concerning collections, the research context of the collections, and cost and expertise involved in repurposing collections to enable sharing. Several solutions to increase sharing are identified. Some have recently been implemented, including enhancing biospecimen locators with tools to guide researchers and facilitating transfer of research collections to centralized biobank infrastructures at the conclusion of projects. New proposed solutions include improving search capabilities within publication databases, and introduction of evidence-based justifications for all new collections into peer-reviewed grant competition processes. It is recognized that there are both scientific factors and practical reasons that can impose limits to sharing biospecimens. However, funding availability, productivity, and progress in health research all stand to benefit from improved sharing of research biospecimen collections.