Discourse particles are crucial elements that subtly shape the meaning of text. These words, often polyfunctional, give rise to nuanced and often quite disparate semantic/discourse effects, as exemplified by the diverse uses of the particle "just" (e.g., exclusive, temporal, emphatic). This work investigates the capacity of LLMs to distinguish the fine-grained senses of English "just", a well-studied example in formal semantics, using data meticulously created and labeled by expert linguists. Our findings reveal that while LLMs exhibit some ability to differentiate between broader categories, they struggle to fully capture more subtle nuances, highlighting a gap in their understanding of discourse particles.