Hypothyroidism refers to the inadequate secretion of thyroid hormones by the thyroid gland in the neck. This is sometimes referred to as having an underactive thyroid. When hypothyroidism is present, there may be changes within the upper airway that lead to difficulties breathing during sleep.
Individuals with hypothyroidism may develop chronic respiratory failure. There are likely multiple factors that lead to this, including a decreased ability to respond to chemical changes within the blood and even damage to the nerves or muscles involved in breathing. Moreover, hypothyroidism may contribute to sleep apnea through enlargement of the tongue (called macroglossia) or disruption of the muscles that control the upper airway.
Fortunately, if hypothyroidism is causing sleep apnea or breathing difficulties, it is improved with thyroid hormone replacement. This is typically taken as a pill called Synthroid (levothyroxine). Therefore, it may be warranted to have your thyroid-stimulating hormone (TSH) checked if you have other symptoms suggesting hypothyroidism.
Skatrud J et al. "Disordered breathing during sleep in hypothyroidism." Am Rev Respir Dis 1981;124:325.