A growing body of literature has focused on detailing the linguistic knowledge embedded in large, pretrained language models. Existing work has shown that non-linguistic biases in models can drive model behavior away from linguistic generalizations. We hypothesized that competing linguistic processes within a language, rather than just non-linguistic model biases, could obscure underlying linguistic knowledge. We tested this claim by exploring a single phenomenon in four languages: English, Chinese, Spanish, and Italian. While human behavior has been found to be similar across languages, we find cross-linguistic variation in model behavior. We show that competing processes in a language act as constraints on model behavior and demonstrate that targeted fine-tuning can re-weight the learned constraints, uncovering otherwise dormant linguistic knowledge in models. Our results suggest that models need to learn both the linguistic constraints in a language and their relative ranking, with mismatches in either producing non-human-like behavior.