Idiomatic expressions are an integral part of human languages, often used to express complex ideas in compressed or conventional ways (e.g. eager beaver as a keen and enthusiastic person). However, their interpretations may not be straightforwardly linked to the meanings of their individual components in isolation and this may have an impact for compositional approaches. In this paper, we investigate to what extent word representation models are able to go beyond compositional word combinations and capture multiword expression idiomaticity and some of the expected properties related to idiomatic meanings. We focus on noun compounds of varying levels of idiomaticity in two languages (English and Portuguese), presenting a dataset of minimal pairs containing human idiomaticity judgments for each noun compound at both type and token levels, their paraphrases and their occurrences in naturalistic and sense-neutral contexts, totalling 32,200 sentences. We propose this set of minimal pairs for evaluating how well a model captures idiomatic meanings, and define a set of fine-grained metrics of Affinity and Scaled Similarity, to determine how sensitive the models are to perturbations that may lead to changes in idiomaticity. The results obtained with a variety of representative and widely used models indicate that, despite superficial indications to the contrary in the form of high similarities, idiomaticity is not yet accurately represented in current models. Moreover, the performance of models with different levels of contextualisation suggests that their ability to capture context is not yet able to go beyond more superficial lexical clues provided by the words and to actually incorporate the relevant semantic clues needed for idiomaticity.