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.

Related Questions

Do I need thyroid hormone if I have low free t3, (liothyronine) normal TSH and high tpo antibodies?

Possibly. The TSH is normal so it might be a variation of normal or euthyroid. It also might be a developing thyroid condition, but there would be symptoms, like fatigue. The tpo antibodies might mean hashimoto's thyroiditis. This is usually associated with hypothyroidism. It would be wise to see your doctor to follow up. Read more...
Maybe. Oral T3 (liothyronine) may be what you need. However, you should consult an endocrinologist. Read more...