Assessments with a large amount of small, similar, or often repetitive tasks are being used in educational, neurocognitive, and psychological contexts. For example, respondents are asked to recognize numbers or letters from a large pool of those and the number of correct answers is a count variable. In 1960, George Rasch developed the Rasch Poisson counts model (RPCM) to handle that type of assessment. This article extends the RPCM into the world of diagnostic classification models (DCMs) where a Poisson distribution is applied to traditional DCMs. A framework of Poisson DCMs is proposed and demonstrated through an operational dataset. This study aims to be exploratory with recommendations for future research given in the end.