This study investigates cross-language lexical competition in the bilingual mental lexicon. It provides evidence for the occurrence of inhibition as well as the commonly reported facilitation during the production of cognates (words with similar phonological form and meaning in two languages) in a mixed picture naming task by highly proficient Welsh-English bilinguals. Previous studies have typically found cognate facilitation. It has previously been proposed (with respect to non-cognates) that cross-language inhibition is limited to low-proficient bilinguals; therefore, we tested highly proficient, early bilinguals. In a mixed naming experiment (i.e., picture naming with language switching), 48 highly proficient, early Welsh-English bilinguals named pictures in Welsh and English, including cognate and non-cognate targets. Participants were English-dominant, Welsh-dominant, or had equal language dominance. The results showed evidence for cognate inhibition in two ways. First, both facilitation and inhibition were found on the cognate trials themselves, compared to non-cognate controls, modulated by the participants' language dominance. The English-dominant group showed cognate inhibition when naming in Welsh (and no difference between cognates and controls when naming in English), and the Welsh-dominant and equal dominance groups generally showed cognate facilitation. Second, cognate inhibition was found as a behavioral adaptation effect, with slower naming for non-cognate filler words in trials after cognates than after non-cognate controls. This effect was consistent across all language dominance groups and both target languages, suggesting that cognate production involved cognitive control even if this was not measurable in the cognate trials themselves. Finally, the results replicated patterns of symmetrical switch costs, as commonly reported for balanced bilinguals. We propose that cognate processing might be affected by two different processes, namely competition at the lexical-semantic level and facilitation at the word form level, and that facilitation at the word form level might (sometimes) outweigh any effects of inhibition at the lemma level. In sum, this study provides evidence that cognate naming can cause costs in addition to benefits. The finding of cognate inhibition, particularly for the highly proficient bilinguals tested, provides strong evidence for the occurrence of lexical competition across languages in the bilingual mental lexicon.