Language processing is not an isolated capacity, but is embedded in other aspects of our cognition. However, it is still largely unexplored to what extent and how language processing interacts with general cognitive resources. This question can be investigated with cognitively constrained computational models, which simulate the cognitive processes involved in language processing. The theoretical claims implemented in cognitive models interact with general architectural constraints such as memory limitations. This way, it generates new predictions that can be tested in experiments, thus generating new data that can give rise to new theoretical insights. This theory-model-experiment cycle is a promising method for investigating aspects of language processing that are difficult to investigate with more traditional experimental techniques. This review specifically examines the language processing models of Lewis and Vasishth (2005), Reitter et al. (2011), and Van Rij et al. (2010), all implemented in the cognitive architecture Adaptive Control of Thought—Rational (Anderson et al., 2004). These models are all limited by the assumptions about cognitive capacities provided by the cognitive architecture, but use different linguistic approaches. Because of this, their comparison provides insight into the extent to which assumptions about general cognitive resources influence concretely implemented models of linguistic competence. For example, the sheer speed and accuracy of human language processing is a current challenge in the field of cognitive modeling, as it does not seem to adhere to the same memory and processing capacities that have been found in other cognitive processes. Architecture-based cognitive models of language processing may be able to make explicit which language-specific resources are needed to acquire and process natural language. The review sheds light on cognitively constrained models of language processing from two angles: we discuss (1) whether currently adopted cognitive assumptions meet the requirements for language processing, and (2) how validated cognitive architectures can constrain linguistically motivated models, which, all other things being equal, will increase the cognitive plausibility of these models. Overall, the evaluation of cognitively constrained models of language processing will allow for a better understanding of the relation between data, linguistic theory, cognitive assumptions, and explanation.