Is it correct? I had a cup of tea for breakfast because I had been running out of coffee.

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Nearly correct, just need to change the end to ' because I had run out of coffee'.

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There are 2 correct possibilities: "I had a cup of tea for breakfast because I ran out of coffee." or "I had a cup of tea for breakfast because I had run out of coffee."

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"because I had run out of coffee." Otherwise it seems good to me.