Human immunodeficiency virus (HIV) infection is currently incurable, due to the persistence of latently infected cells. The 'shock and kill' approach to a cure proposes to eliminate this reservoir via transcriptional activation of latent proviruses, enabling direct or indirect killing of infected cells. Currently available latency-reversing agents (LRAs) have however proven ineffective. To understand why, we used a novel HIV reporter strain in primary CD4⁺ T cells and determined which latently infected cells are reactivatable by current candidate LRAs. Remarkably, none of these agents reactivated more than 5% of cells carrying a latent provirus. Sequencing analysis of reactivatable vs. non-reactivatable populations revealed that the integration sites were distinguishable in terms of chromatin functional states. Our findings challenge the feasibility of 'shock and kill', and suggest the need to explore other strategies to control the latent HIV reservoir.