How accurate is a fine neddle biospy on the thyroid?

Good--but limited. Fna is an excellent technique for papillary, medullary and undifferentiated cancer. However, there are limitations for a follicular tumor or nodule. You need to look at the periphery of a nodule to see if there is invasion before you can accurately classify these lesions. So, if you have a single nodule, fna is a good screening tool, but it cannot absolutely access all types of thyroid conditions.
Very accurate. Thyroid fine needle aspiration (fna) is done with a thin needle, usually under ultrasound guidance. Done that way, fna will identify 94% of abnormal lesions in the thyroid, with a specificity of 81%. There is no better test short of doing surgery. Fna is usually used to decide on whether surgery is necessary. Since there is a low false negative rate, follow up ultrasound is often recommended.
Can be very accurate. In the best of hands, thyroid fna can detect >95% of cancers. The diagnostic rate and sensitivity varies greatly (miss rate can be as high as 50%). High risk lesions (benign adenoma and follicular cancer), can be easily distinguished from common benign nodules by skilled cytologists. It is appropriate to ask about non-diagnostic /indeterminate rates. Search "quality improvement in thyroid fna".