Can you have both hyper and hypo thyroidism?

Not simultaneously. Hi. Yes, you can, but not simultaneously. For example, treatment of Graves' disease often renders the patients hypothyroid. Hypothyroidism can sequentially follow hyperthyroidism. Same for a self limited process called a subacute thyroiditis - hyperthyroidism followed by hypothyroidism. Hypothyroidism can be over treated causing hyperthyroidism. Hyper and hypo cannot be present at the same time.