Yes, it is possible to have pieces of cork in your bottle - even in a bottle which has never been opened. How long ago did you purchase it and from whom? If you haven't had it that long, let the person know who sold it to you. You said it's a 30 yo, but when was it bottled? And to answer your question - yes, pieces of cork can affect the taste of the Scotch.
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Jojo 
Whisky.com
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