Is phenomenal consciousness constitutively related to cognitive access? Despite being a fundamental issue for any science of consciousness, its empirical study faces a severe methodological puzzle. Recent years have seen numerous attempts to address this puzzle, either in practice, by offering evidence for a positive or negative answer, or in principle, by proposing a framework for eventual resolution. The present paper critically considers these endeavours, including partial-report, metacognitive and no-report paradigms, as well as the theoretical proposal that we can make progress by studying phenomenal consciousness as a natural kind. It is argued that the methodological puzzle remains obdurately with us and that, for now, we must adopt an attitude of humility towards the phenomenal. This article is part of the theme issue ‘Perceptual consciousness and cognitive access’.