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.