This paper identifies two opposing factors within the overall trend towards increased accessibility of museum collections on the Web: the need to make museum Web sites more attractive to non-traditional audiences and the need to ensure that accessibility is maximised for those with disabilities. The rapidly increasing availability of broadband creates opportunities for museums to create rich multimedia Web sites that can attract much wider audiences. Without careful design such sites can also reduce the diversity of potential users by creating barriers to access for disabled users. By analysing two recent award-winning sites, this paper shows how rich multimedia content can be designed to enhance accessibility, and, conversely, that 'text-only' sites may reduce the accessibility of a site while unnecessarily increasing costs. We conclude that in order to 'square the triangle' of diversity, access and accessibility, the solution lies in thinking in terms of improvements in overall usability, rather than simply responding to the concept of 'accessibility' as it is narrowly defined by laws and official guidelines. However, our ability to apply this thinking is constrained by the limitations of current accessibility assessment tools such as Bobby. We describe a reduced but key set of accessibility checks, suggest a set of heuristics for undertaking them, and illustrate their application upon the sites in question.