Concepts and their mental associations influence how language is processed and used. Networks represent powerful models for exploring such cognitive system, known as mental lexicon. This study investigates lexicon robustness to progressive word failure with multiplex network attacks. The average lexicon of an adult English speaker is built by considering 16000 words connected through semantic free associations and phonological sound similarities. Progressive structural degradation is modelled as random and targeted attacks. Words with higher psycholinguistic features (e.g. frequency, length, age of acquisition, polysemy) or network centrality (e.g. closeness, PageRank, betweenness and degree) are targeted first. Aphasia-inspired attacks are introduced here and target first words named correctly, more or less frequently, by patients with anomic aphasia, a pathology disrupting word finding. Robustness is measured as connectedness, fundamental for activation spreading and lexical retrieval, and viability, a multi-layer connectivity identifying language kernels. The lexicon is resilient to random, aphasia-inspired and psycholinguistic attacks. Catastrophic phase transitions happen when phonological and semantic degrees are combined, making the lexicon fragile to multidegree attacks. The viable kernel is fragile to multi-PageRank and to aphasia-inspired attacks. Consequently, connectedness in the lexicon is mediated by hubs, whereas viability enables a lexical semantic/phonological interplay and corresponds to a facilitative naming effect in aphasia. These effects persist also through ageing, in different network representations of younger and older lexicons. This study indicates the need to prevent failure of high multidegree and viable words in the mental lexicon when pursuing the design of effective language restoration strategies against cognitive impairing.