The growth in prominence of large language models (LLMs) in everyday life can be largely attributed to their generative abilities, yet some of this is also owed to the risks and costs associated with their use. On one front is their tendency to \textit{hallucinate} false or misleading information, limiting their reliability. On another is the increasing focus on the computational limitations associated with traditional self-attention based LLMs, which has brought about new alternatives, in particular recurrent models, meant to overcome them. Yet it remains uncommon to consider these two concerns simultaneously. Do changes in architecture exacerbate/alleviate existing concerns about hallucinations? Do they affect how and where they occur? Through an extensive evaluation, we study how these architecture-based inductive biases affect the propensity to hallucinate. While hallucination remains a general phenomenon not limited to specific architectures, the situations in which they occur and the ease with which specific types of hallucinations can be induced can significantly differ based on the model architecture. These findings highlight the need for better understanding both these problems in conjunction with each other, as well as consider how to design more universal techniques for handling hallucinations.