The answer is yes or no. Under normal circumstances and everyday wear, a hat is not going to cause hair loss. Can accessories like helmets worsen hair loss? Accessories, hats like a motorbike crash helmet may be tight but the time used is minimal to cause actual hair loss. Traction can be caused by pulling the hair from the root but this is generally minimal. Tight headbands and clips can also cause the hair to break.
Head scarfs worn constantly especially when tight against the skin can cause traction alopecia. If worn for long periods of time when removed often show areas of hair loss. For ladies, different hairstyles using bands or braids need to be tied with care. Avoid tying your hair too tightly.
Often if the signs are already visible the best can be to stop the problem increasing. Treatment can help to strengthen the hair but may not encourage hair to regrow again. The quality of the skin and the length of the condition are factors in the healing process. A personal consultation can be the first step if you are concerned about your hair growth.