Can you have a thyroid problem if you have a normal thyroid-stimulating hormone (tsh)?

Yes: The 'normal' TSH level is determined statistically, not clinically. Some people in the upper part of the 'normal' range will have symptoms and will benefit from treatment. This is similar to people on thyroid hormone treatment who have an increase in tsh, due to further failure of the thyroid, report recurence of symptoms and improvement with adjustment of their treatment.

Yes: A normal TSH doesn't rule out hypothyoidism as a result of pituitary hypofunction, nor does it rule out thyroiditis, nodular disease, etc.