HIV latency is a major obstacle to curing infection. Current strategies to eradicate HIV aim at increasing transcription of the latent provirus. In the present study we observed that latently infected CD4+ T cells from HIV-infected individuals failed to produce viral particles upon ex vivo exposure to SAHA (vorinostat), despite effective inhibition of histone deacetylases. To identify steps that were not susceptible to the action of SAHA or other latency reverting agents, we used a primary CD4+ T cell model, joint host and viral RNA sequencing, and a viral-encoded reporter. This model served to investigate the characteristics of latently infected cells, the dynamics of HIV latency, and the process of reactivation induced by various stimuli. During latency, we observed persistence of viral transcripts but only limited viral translation. Similarly, the reactivating agents SAHA and disulfiram successfully increased viral transcription, but failed to effectively enhance viral translation, mirroring the ex vivo data. This study highlights the importance of post-transcriptional blocks as one mechanism leading to HIV latency that needs to be relieved in order to purge the viral reservoir.