Tim Berners-Lee claimed in 2001 that “the power of the Web is in its universality. Access by everyone regardless of disability is an essential aspect”. A considerable amount of work has been done to make the web accessible to those with sensory or motor disability, with an increasing number of government and enterprise intranet webs being “accessible”, and also with some consortiums and groups seriously approaching this commitment. Some authors, such as Harrysson, have already highlighted the need for a cognitively accessible web. However, in spite of good intentions, there has been little work to date that has tackled this task. At least until now, the existing WAI and NI4 recommendations about cognitive disability are extremely difficult (if not impossible) to test, as they are only general recommendations. This article explains an alternative Web that has been constructed and tested on a sample of participants with cognitive disabilities (N = 20) with positive results encouraging us to dedicate more effort to fine tune their requirements regarding specific cognitive deficits and automating the process of creating and testing cognitively accessible web content. This alternative web implies the use of a simplified web browser and an adequate web design. Discussion of the need to have several levels of cognitive accessibility, equivalent (although not identical) content for this collective and the need for testable protocols of accessibility that support these people's needs is also included. This article finishes with conclusions about the potential impact of accessible pages in the daily life of people suffering from cognitive deficits, outlining the features to be considered within a user profile specification that support cognitive difficulties and with reflections about the suitability of Semantic Web Technologies for future developments in this field.