Klinefelter syndrome (countable)
He has got a Klinefelter syndrome.
She has got a Down syndrome.
mosja got a These are not needed.
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mosjaKlinefelter syndrome (countable)
He has got a Klinefelter syndrome.
She has got a Down syndrome.
The word 'syndrome' by itself is countable, but specific syndromes are not countable unless more than one syndrome has been given the same name, and I don't think there are any syndromes like that.
Klinefelte